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In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.
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https://en.wikipedia.org/wiki/Locally_finite_group
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In mathematics, in the field of group theory, a metanilpotent group is a group that is nilpotent by nilpotent. In other words, it has a normal nilpotent subgroup such that the quotient group is also nilpotent. In symbols, G {\displaystyle G} is metanilpotent if there is a normal subgroup N {\displaystyle N} such that both N {\displaystyle N} and G / N {\displaystyle G/N} are nilpotent. The following are clear: Every metanilpotent group is a solvable group. Every subgroup and every quotient of a metanilpotent group is metanilpotent.
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https://en.wikipedia.org/wiki/Metanilpotent_group
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In mathematics, in the field of group theory, a modular subgroup is a subgroup that is a modular element in the lattice of subgroups, where the meet operation is defined by the intersection and the join operation is defined by the subgroup generated by the union of subgroups. By the modular property of groups, every quasinormal subgroup (that is, a subgroup that permutes with all subgroups) is modular. In particular, every normal subgroup is modular.
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https://en.wikipedia.org/wiki/Modular_subgroup
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In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it within that subgroup. In symbols, H {\displaystyle H} is paranormal in G {\displaystyle G} if given any g {\displaystyle g} in G {\displaystyle G} , the subgroup K {\displaystyle K} generated by H {\displaystyle H} and H g {\displaystyle H^{g}} is also equal to H K {\displaystyle H^{K}} . Equivalently, a subgroup is paranormal if its weak closure and normal closure coincide in all intermediate subgroups. Here are some facts relating paranormality to other subgroup properties: Every pronormal subgroup, and hence, every normal subgroup and every abnormal subgroup, is paranormal. Every paranormal subgroup is a polynormal subgroup. In finite solvable groups, every polynormal subgroup is paranormal.
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https://en.wikipedia.org/wiki/Paranormal_subgroup
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In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term quasinormal subgroup was introduced by Øystein Ore in 1937. Two subgroups are said to permute (or commute) if any element from the first subgroup, times an element of the second subgroup, can be written as an element of the second subgroup, times an element of the first subgroup.
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https://en.wikipedia.org/wiki/PT-group
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That is, H {\displaystyle H} and K {\displaystyle K} as subgroups of G {\displaystyle G} are said to commute if HK = KH, that is, any element of the form h k {\displaystyle hk} with h ∈ H {\displaystyle h\in H} and k ∈ K {\displaystyle k\in K} can be written in the form k ′ h ′ {\displaystyle k'h'} where k ′ ∈ K {\displaystyle k'\in K} and h ′ ∈ H {\displaystyle h'\in H} . Every normal subgroup is quasinormal, because a normal subgroup commutes with every element of the group. The converse is not true.
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https://en.wikipedia.org/wiki/PT-group
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For instance, any extension of a cyclic p {\displaystyle p} -group by another cyclic p {\displaystyle p} -group for the same (odd) prime has the property that all its subgroups are quasinormal. However, not all of its subgroups need be normal. Every quasinormal subgroup is a modular subgroup, that is, a modular element in the lattice of subgroups.
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https://en.wikipedia.org/wiki/PT-group
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This follows from the modular property of groups. If all subgroups are quasinormal, then the group is called an Iwasawa group—sometimes also called a modular group, although this latter term has other meanings. In any group, every quasinormal subgroup is ascendant. A conjugate permutable subgroup is one that commutes with all its conjugate subgroups. Every quasinormal subgroup is conjugate permutable.
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https://en.wikipedia.org/wiki/PT-group
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In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G. In notation, H {\displaystyle H} is k {\displaystyle k} -subnormal in G {\displaystyle G} if there are subgroups H = H 0 , H 1 , H 2 , … , H k = G {\displaystyle H=H_{0},H_{1},H_{2},\ldots ,H_{k}=G} of G {\displaystyle G} such that H i {\displaystyle H_{i}} is normal in H i + 1 {\displaystyle H_{i+1}} for each i {\displaystyle i} . A subnormal subgroup is a subgroup that is k {\displaystyle k} -subnormal for some positive integer k {\displaystyle k} . Some facts about subnormal subgroups: A 1-subnormal subgroup is a proper normal subgroup (and vice versa). A finitely generated group is nilpotent if and only if each of its subgroups is subnormal.
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https://en.wikipedia.org/wiki/Subnormal_subgroup
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Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal. Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal.
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https://en.wikipedia.org/wiki/Subnormal_subgroup
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Every 2-subnormal subgroup is a conjugate-permutable subgroup.The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality. If every subnormal subgroup of G is normal in G, then G is called a T-group.
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https://en.wikipedia.org/wiki/Subnormal_subgroup
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In mathematics, in the field of group theory, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is called c-normal if there is a normal subgroup T {\displaystyle T} of G {\displaystyle G} such that H T = G {\displaystyle HT=G} and the intersection of H {\displaystyle H} and T {\displaystyle T} lies inside the normal core of H {\displaystyle H} . For a weakly c-normal subgroup, we only require T {\displaystyle T} to be subnormal. Here are some facts on c-normal subgroups: Every normal subgroup is c-normal Every retract is c-normal Every c-normal subgroup is weakly c-normal
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https://en.wikipedia.org/wiki/C-normal_subgroup
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In mathematics, in the field of group theory, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is said to be weakly normal if whenever H g ≤ N G ( H ) {\displaystyle H^{g}\leq N_{G}(H)} , we have g ∈ N G ( H ) {\displaystyle g\in N_{G}(H)} . Every pronormal subgroup is weakly normal.
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https://en.wikipedia.org/wiki/Weakly_normal_subgroup
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In mathematics, in the field of group theory, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is termed malnormal if for any x {\displaystyle x} in G {\displaystyle G} but not in H {\displaystyle H} , H {\displaystyle H} and x H x − 1 {\displaystyle xHx^{-1}} intersect in the identity element.Some facts about malnormality: An intersection of malnormal subgroups is malnormal. Malnormality is transitive, that is, a malnormal subgroup of a malnormal subgroup is malnormal. The trivial subgroup and the whole group are malnormal subgroups. A normal subgroup that is also malnormal must be one of these.
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https://en.wikipedia.org/wiki/Malnormal_subgroup
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Every malnormal subgroup is a special type of C-group called a trivial intersection subgroup or TI subgroup.When G is finite, a malnormal subgroup H distinct from 1 and G is called a "Frobenius complement". The set N of elements of G which are, either equal to 1, or non-conjugate to any element of H, is a normal subgroup of G, called the "Frobenius kernel", and G is the semi-direct product of H and N (Frobenius' theorem). == References ==
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https://en.wikipedia.org/wiki/Malnormal_subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup is subnormal.
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https://en.wikipedia.org/wiki/Ascendant_subgroup
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Here are some properties of ascendant subgroups: Every subnormal subgroup is ascendant; every ascendant subgroup is serial. In a finite group, the properties of being ascendant and subnormal are equivalent. An arbitrary intersection of ascendant subgroups is ascendant. Given any subgroup, there is a minimal ascendant subgroup containing it.
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https://en.wikipedia.org/wiki/Ascendant_subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to be conjugacy-closed if any two elements of the subgroup that are conjugate in the group are also conjugate in the subgroup. An alternative characterization of conjugacy-closed normal subgroups is that all class automorphisms of the whole group restrict to class automorphisms of the subgroup. The following facts are true regarding conjugacy-closed subgroups: Every central factor (a subgroup that may occur as a factor in some central product) is a conjugacy-closed subgroup. Every conjugacy-closed normal subgroup is a transitively normal subgroup.
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https://en.wikipedia.org/wiki/Conjugacy-closed_subgroup
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The property of being conjugacy-closed is transitive, that is, every conjugacy-closed subgroup of a conjugacy-closed subgroup is conjugacy-closed.The property of being conjugacy-closed is sometimes also termed as being conjugacy stable. It is a known result that for finite field extensions, the general linear group of the base field is a conjugacy-closed subgroup of the general linear group over the extension field. This result is typically referred to as a stability theorem. A subgroup is said to be strongly conjugacy-closed if all intermediate subgroups are also conjugacy-closed.
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https://en.wikipedia.org/wiki/Conjugacy-closed_subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor. The series may be infinite. If the series is finite, then the subgroup is subnormal.
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https://en.wikipedia.org/wiki/Descendant_subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to be fully normalized if every automorphism of the subgroup lifts to an inner automorphism of the whole group. Another way of putting this is that the natural embedding from the Weyl group of the subgroup to its automorphism group is surjective. In symbols, a subgroup H {\displaystyle H} is fully normalized in G {\displaystyle G} if, given an automorphism σ {\displaystyle \sigma } of H {\displaystyle H} , there is a g ∈ G {\displaystyle g\in G} such that the map x ↦ g x g − 1 {\displaystyle x\mapsto gxg^{-1}} , when restricted to H {\displaystyle H} is equal to σ {\displaystyle \sigma } . Some facts: Every group can be embedded as a normal and fully normalized subgroup of a bigger group.
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https://en.wikipedia.org/wiki/Fully_normalized_subgroup
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A natural construction for this is the holomorph, which is its semidirect product with its automorphism group. A complete group is fully normalized in any bigger group in which it is embedded because every automorphism of it is inner. Every fully normalized subgroup has the automorphism extension property.
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https://en.wikipedia.org/wiki/Fully_normalized_subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to be polynormal if its closure under conjugation by any element of the group can also be achieved via closure by conjugation by some element in the subgroup generated. In symbols, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is called polynormal if for any g ∈ G {\displaystyle g\in G} the subgroup K = H < g > {\displaystyle K=H^{ }} is the same as H H < g > {\displaystyle H^{H^{ }}} . Here are the relationships with other subgroup properties: Every weakly pronormal subgroup is polynormal. Every paranormal subgroup is polynormal. == References ==
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https://en.wikipedia.org/wiki/Polynormal_subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to be transitively normal in the group if every normal subgroup of the subgroup is also normal in the whole group. In symbols, H {\displaystyle H} is a transitively normal subgroup of G {\displaystyle G} if for every K {\displaystyle K} normal in H {\displaystyle H} , we have that K {\displaystyle K} is normal in G {\displaystyle G} .An alternate way to characterize these subgroups is: every normal subgroup preserving automorphism of the whole group must restrict to a normal subgroup preserving automorphism of the subgroup. Here are some facts about transitively normal subgroups: Every normal subgroup of a transitively normal subgroup is normal. Every direct factor, or more generally, every central factor is transitively normal.
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https://en.wikipedia.org/wiki/Transitively_normal_subgroup
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Thus, every central subgroup is transitively normal. A transitively normal subgroup of a transitively normal subgroup is transitively normal. A transitively normal subgroup is normal.
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https://en.wikipedia.org/wiki/Transitively_normal_subgroup
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In mathematics, in the field of group theory, a subgroup of a group is said to have the Congruence Extension Property or to be a CEP subgroup if every congruence on the subgroup lifts to a congruence of the whole group. Equivalently, every normal subgroup of the subgroup arises as the intersection with the subgroup of a normal subgroup of the whole group. In symbols, a subgroup H {\displaystyle H} is a CEP subgroup in a group G {\displaystyle G} if every normal subgroup N {\displaystyle N} of H {\displaystyle H} can be realized as H ∩ M {\displaystyle H\cap M} where M {\displaystyle M} is normal in G {\displaystyle G} . The following facts are known about CEP subgroups: Every retract has the CEP. Every transitively normal subgroup has the CEP.
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https://en.wikipedia.org/wiki/CEP_subgroup
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In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is identity on the subgroup. In symbols, H {\displaystyle H} is a retract of G {\displaystyle G} if and only if there is an endomorphism σ: G → G {\displaystyle \sigma :G\to G} such that σ ( h ) = h {\displaystyle \sigma (h)=h} for all h ∈ H {\displaystyle h\in H} and σ ( g ) ∈ H {\displaystyle \sigma (g)\in H} for all g ∈ G {\displaystyle g\in G} .The endomorphism itself (having this property) is an idempotent element in the transformation monoid of endomorphisms, so it is called an idempotent endomorphism or a retraction.The following is known about retracts: A subgroup is a retract if and only if it has a normal complement. The normal complement, specifically, is the kernel of the retraction.
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https://en.wikipedia.org/wiki/Retract_(group_theory)
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Every direct factor is a retract. Conversely, any retract which is a normal subgroup is a direct factor. Every retract has the congruence extension property. Every regular factor, and in particular, every free factor, is a retract.
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https://en.wikipedia.org/wiki/Retract_(group_theory)
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In mathematics, in the field of group theory, a subgroup of a group is termed central if it lies inside the center of the group. Given a group G {\displaystyle G} , the center of G {\displaystyle G} , denoted as Z ( G ) {\displaystyle Z(G)} , is defined as the set of those elements of the group which commute with every element of the group. The center is a characteristic subgroup. A subgroup H {\displaystyle H} of G {\displaystyle G} is termed central if H ≤ Z ( G ) {\displaystyle H\leq Z(G)} .
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https://en.wikipedia.org/wiki/Central_subgroup
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Central subgroups have the following properties: They are abelian groups (because, in particular, all elements of the center must commute with each other). They are normal subgroups. They are central factors, and are hence transitively normal subgroups.
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https://en.wikipedia.org/wiki/Central_subgroup
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In mathematics, in the field of group theory, an FC-group is a group in which every conjugacy class of elements has finite cardinality. The following are some facts about FC-groups: Every finite group is an FC-group. Every abelian group is an FC-group. The following property is stronger than the property of being FC: every subgroup has finite index in its normal closure.
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https://en.wikipedia.org/wiki/FC-group
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In mathematics, in the field of group theory, especially in the study of p-groups and pro-p-groups, the concept of powerful p-groups plays an important role. They were introduced in (Lubotzky & Mann 1987), where a number of applications are given, including results on Schur multipliers. Powerful p-groups are used in the study of automorphisms of p-groups (Khukhro 1998), the solution of the restricted Burnside problem (Vaughan-Lee 1993), the classification of finite p-groups via the coclass conjectures (Leedham-Green & McKay 2002), and provided an excellent method of understanding analytic pro-p-groups (Dixon et al. 1991).
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https://en.wikipedia.org/wiki/Powerful_p-group
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In mathematics, in the field of group theory, the norm of a group is the intersection of the normalizers of all its subgroups. This is also termed the Baer norm, after Reinhold Baer. The following facts are true for the Baer norm: It is a characteristic subgroup.
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https://en.wikipedia.org/wiki/Norm_(group)
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It contains the center of the group. It is contained inside the second term of the upper central series. It is a Dedekind group, so is either abelian or has a direct factor isomorphic to the quaternion group. If it contains an element of infinite order, then it is equal to the center of the group.
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https://en.wikipedia.org/wiki/Norm_(group)
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In mathematics, in the field of group theory, the perfect core (or perfect radical) of a group is its largest perfect subgroup. Its existence is guaranteed by the fact that the subgroup generated by a family of perfect subgroups is again a perfect subgroup. The perfect core is also the point where the transfinite derived series stabilizes for any group. A group whose perfect core is trivial is termed a hypoabelian group.
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https://en.wikipedia.org/wiki/Perfect_core
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Every solvable group is hypoabelian, and so is every free group. More generally, every residually solvable group is hypoabelian. The quotient of a group G by its perfect core is hypoabelian, and is called the hypoabelianization of G. == References ==
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https://en.wikipedia.org/wiki/Perfect_core
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In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form T λ u ( x ) = ∫ R n e i λ S ( x , y ) a ( x , y ) u ( y ) d y , x ∈ R m , y ∈ R n , {\displaystyle T_{\lambda }u(x)=\int _{\mathbb {R} ^{n}}e^{i\lambda S(x,y)}a(x,y)u(y)\,dy,\qquad x\in \mathbb {R} ^{m},\quad y\in \mathbb {R} ^{n},} where the function S(x,y) is called the phase of the operator and the function a(x,y) is called the symbol of the operator. λ is a parameter. One often considers S(x,y) to be real-valued and smooth, and a(x,y) smooth and compactly supported. Usually one is interested in the behavior of Tλ for large values of λ. Oscillatory integral operators often appear in many fields of mathematics (analysis, partial differential equations, integral geometry, number theory) and in physics. Properties of oscillatory integral operators have been studied by Elias Stein and his school.
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https://en.wikipedia.org/wiki/Oscillatory_integral_operator
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In mathematics, in the field of harmonic analysis, the van der Corput lemma is an estimate for oscillatory integrals named after the Dutch mathematician J. G. van der Corput. The following result is stated by E. Stein:Suppose that a real-valued function ϕ ( x ) {\displaystyle \phi (x)} is smooth in an open interval ( a , b ) {\displaystyle (a,b)} , and that | ϕ ( k ) ( x ) | ≥ 1 {\displaystyle |\phi ^{(k)}(x)|\geq 1} for all x ∈ ( a , b ) {\displaystyle x\in (a,b)} . Assume that either k ≥ 2 {\displaystyle k\geq 2} , or that k = 1 {\displaystyle k=1} and ϕ ′ ( x ) {\displaystyle \phi '(x)} is monotone for x ∈ R {\displaystyle x\in \mathbb {R} } . Then there is a constant c k {\displaystyle c_{k}} , which does not depend on ϕ {\displaystyle \phi } , such that | ∫ a b e i λ ϕ ( x ) | ≤ c k λ − 1 / k {\displaystyle {\bigg |}\int _{a}^{b}e^{i\lambda \phi (x)}{\bigg |}\leq c_{k}\lambda ^{-1/k}} for any λ ∈ R {\displaystyle \lambda \in \mathbb {R} } .
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https://en.wikipedia.org/wiki/Van_der_Corput_lemma_(harmonic_analysis)
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In mathematics, in the field of homological algebra, given an abelian category C {\displaystyle {\mathcal {C}}} having enough injectives and an additive (covariant) functor F: C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} ,an acyclic object with respect to F {\displaystyle F} , or simply an F {\displaystyle F} -acyclic object, is an object A {\displaystyle A} in C {\displaystyle {\mathcal {C}}} such that R i F ( A ) = 0 {\displaystyle {\rm {R}}^{i}F(A)=0\,\!} for all i > 0 {\displaystyle i>0\,\!} ,where R i F {\displaystyle {\rm {R}}^{i}F} are the right derived functors of F {\displaystyle F} .
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https://en.wikipedia.org/wiki/Acyclic_object
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In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors G ∘ F {\displaystyle G\circ F} , from knowledge of the derived functors of F {\displaystyle F} and G {\displaystyle G} . Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.
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https://en.wikipedia.org/wiki/Grothendieck_spectral_sequence
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In mathematics, in the field of measure theory, τ-additivity is a certain property of measures on topological spaces. A measure or set function μ {\displaystyle \mu } on a space X {\displaystyle X} whose domain is a sigma-algebra Σ {\displaystyle \Sigma } is said to be τ-additive if for any upward-directed family G ⊆ Σ {\displaystyle {\mathcal {G}}\subseteq \Sigma } of nonempty open sets such that its union is in Σ , {\displaystyle \Sigma ,} the measure of the union is the supremum of measures of elements of G ; {\displaystyle {\mathcal {G}};} that is,:
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https://en.wikipedia.org/wiki/Τ-additivity
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In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent. The equation is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum Hamming distance 5 or 6.
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https://en.wikipedia.org/wiki/Ramanujan–Nagell_equation
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In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such an equation the zeros of the two solutions are alternating.
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https://en.wikipedia.org/wiki/Sturm_separation_theorem
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In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation F ( x , y , y ′ , … , y ( n − 1 ) ) = y ( n ) x ∈ [ 0 , + ∞ ) {\displaystyle F(x,y,y',\ \dots ,\ y^{(n-1)})=y^{(n)}\quad x\in [0,+\infty )} is called oscillating if it has an infinite number of roots; otherwise it is called non-oscillating. The differential equation is called oscillating if it has an oscillating solution. The number of roots carries also information on the spectrum of associated boundary value problems.
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https://en.wikipedia.org/wiki/Oscillation_theory
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In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real domain. Let pi, qi for i = 1, 2 be real-valued continuous functions on the interval and let ( p 1 ( x ) y ′ ) ′ + q 1 ( x ) y = 0 {\displaystyle (p_{1}(x)y^{\prime })^{\prime }+q_{1}(x)y=0} ( p 2 ( x ) y ′ ) ′ + q 2 ( x ) y = 0 {\displaystyle (p_{2}(x)y^{\prime })^{\prime }+q_{2}(x)y=0} be two homogeneous linear second order differential equations in self-adjoint form with 0 < p 2 ( x ) ≤ p 1 ( x ) {\displaystyle 0
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https://en.wikipedia.org/wiki/Sturm-Picone_comparison_theorem
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In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions.
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https://en.wikipedia.org/wiki/Volkenborn_integral
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In mathematics, in the field of potential theory, the fine topology is a natural topology for setting the study of subharmonic functions. In the earliest studies of subharmonic functions, namely those for which Δ u ≥ 0 , {\displaystyle \Delta u\geq 0,} where Δ {\displaystyle \Delta } is the Laplacian, only smooth functions were considered. In that case it was natural to consider only the Euclidean topology, but with the advent of upper semi-continuous subharmonic functions introduced by F. Riesz, the fine topology became the more natural tool in many situations.
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https://en.wikipedia.org/wiki/Classical_fine_topology
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In mathematics, in the field of ring theory, a lattice is a module over a ring which is embedded in a vector space over a field, giving an algebraic generalisation of the way a lattice group is embedded in a real vector space.
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https://en.wikipedia.org/wiki/Lattice_(module)
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In mathematics, in the field of topology, a topological space X {\displaystyle X} is said to be collectionwise Hausdorff if given any closed discrete subset of X {\displaystyle X} , there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.Here a subset S ⊆ X {\displaystyle S\subseteq X} being discrete has the usual meaning of being a discrete space with the subspace topology (i.e., all points of S {\displaystyle S} are isolated in S {\displaystyle S} ).
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https://en.wikipedia.org/wiki/Collectionwise_Hausdorff_space
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In mathematics, in the field of topology, a topological space is called supercompact if there is a subbasis such that every open cover of the topological space from elements of the subbasis has a subcover with at most two subbasis elements. Supercompactness and the related notion of superextension was introduced by J. de Groot in 1967.
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https://en.wikipedia.org/wiki/Supercompact_space
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In mathematics, in the field of topology, a topological space is said to be a Volterra space if any finite intersection of dense Gδ subsets is dense. Every Baire space is Volterra, but the converse is not true. In fact, any metrizable Volterra space is Baire. The name refers to a paper of Vito Volterra in which he uses the fact that (in modern notation) the intersection of two dense G-delta sets in the real numbers is again dense.
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https://en.wikipedia.org/wiki/Volterra_space
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In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the same indexing set, with the property that the closure of each open set in the shrinking lies inside the corresponding original open set.
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https://en.wikipedia.org/wiki/Shrinking_space
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In mathematics, in the field of topology, a topological space is said to be a-paracompact if every open cover of the space has a locally finite refinement. In contrast to the definition of paracompactness, the refinement is not required to be open. Every paracompact space is a-paracompact, and in regular spaces the two notions coincide.
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https://en.wikipedia.org/wiki/A-paracompact_space
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In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence. Clearly, this forces the union of the sequence to be the whole space, because every point is compact and hence must lie in one of the compact sets.
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https://en.wikipedia.org/wiki/Hemicompact_space
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In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has a neighbourhood that is a Hausdorff space under the subspace topology.
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https://en.wikipedia.org/wiki/Locally_Hausdorff_space
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In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of pseudocompactness. Pseudocompact spaces were defined by Edwin Hewitt in 1948.
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https://en.wikipedia.org/wiki/Pseudocompact_space
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In mathematics, in the field of topology, a topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Note the following: Every normal space is pseudonormal. Every pseudonormal space is regular.An example of a pseudonormal Moore space that is not metrizable was given by F. B. Jones (1937), in connection with the conjecture that all normal Moore spaces are metrizable. == References ==
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https://en.wikipedia.org/wiki/Pseudonormal_space
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In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regular Hausdorff and it contains every point of its Stone–Čech compactification which is real (meaning that the quotient field at that point of the ring of real functions is the reals). Realcompact spaces have also been called Q-spaces, saturated spaces, functionally complete spaces, real-complete spaces, replete spaces and Hewitt–Nachbin spaces (named after Edwin Hewitt and Leopoldo Nachbin). Realcompact spaces were introduced by Hewitt (1948).
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https://en.wikipedia.org/wiki/Realcompact_space
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In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary algebraic operations on real numbers, and then take the logarithm to reverse the initial exponentiation. Such operations are also known as, e.g., logarithmic addition, etc. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base b for the exponent and logarithm (b is a choice of logarithmic unit), which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a base b < 1 is equivalent to using a negative sign and using the inverse 1/b > 1.
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https://en.wikipedia.org/wiki/Log_semiring
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If not qualified, the base is conventionally taken to be e or 1/e, which corresponds to e with a negative. The log semiring has the tropical semiring as limit ("tropicalization", "dequantization") as the base goes to infinity b → ∞ {\displaystyle b\to \infty } (max-plus semiring) or to zero b → 0 {\displaystyle b\to 0} (min-plus semiring), and thus can be viewed as a deformation ("quantization") of the tropical semiring.
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https://en.wikipedia.org/wiki/Log_semiring
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Notably, the addition operation, logadd (for multiple terms, LogSumExp) can be viewed as a deformation of maximum or minimum. The log semiring has applications in mathematical optimization, since it replaces the non-smooth maximum and minimum by a smooth operation. The log semiring also arises when working with numbers that are logarithms (measured on a logarithmic scale), such as decibels (see Decibel § Addition), log probability, or log-likelihoods.
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https://en.wikipedia.org/wiki/Log_semiring
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In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor A {\displaystyle \mathbf {A} } are the coefficients of the characteristic polynomial p ( λ ) = det ( A − λ I ) {\displaystyle \ p(\lambda )=\det(\mathbf {A} -\lambda \mathbf {I} )} ,where I {\displaystyle \mathbf {I} } is the identity operator and λ i ∈ C {\displaystyle \lambda _{i}\in \mathbb {C} } represent the polynomial's eigenvalues. More broadly, any scalar-valued function f ( A ) {\displaystyle f(\mathbf {A} )} is an invariant of A {\displaystyle \mathbf {A} } if and only if f ( Q A Q T ) = f ( A ) {\displaystyle f(\mathbf {Q} \mathbf {A} \mathbf {Q} ^{T})=f(\mathbf {A} )} for all orthogonal Q {\displaystyle \mathbf {Q} } . This means that a formula expressing an invariant in terms of components, A i j {\displaystyle A_{ij}} , will give the same result for all Cartesian bases. For example, even though individual diagonal components of A {\displaystyle \mathbf {A} } will change with a change in basis, the sum of diagonal components will not change.
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https://en.wikipedia.org/wiki/Invariants_of_tensors
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In mathematics, in the framework of one-universe foundation for category theory, the term "conglomerate" is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of a Grothendieck universe.
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https://en.wikipedia.org/wiki/Conglomerate_(mathematics)
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In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit. Consider the continuous dynamical system described by the ordinary differential equation Suppose there are equilibria at x = x 0 , x 1 . {\displaystyle x=x_{0},x_{1}.} Then a solution ϕ ( t ) {\displaystyle \phi (t)} is a heteroclinic orbit from x 0 {\displaystyle x_{0}} to x 1 {\displaystyle x_{1}} if both limits are satisfied: This implies that the orbit is contained in the stable manifold of x 1 {\displaystyle x_{1}} and the unstable manifold of x 0 {\displaystyle x_{0}} .
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https://en.wikipedia.org/wiki/Heteroclinic_orbit
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In mathematics, in the realm of abelian group theory, a group is said to be algebraically compact if it is a direct summand of every abelian group containing it as a pure subgroup. Equivalent characterizations of algebraic compactness: The reduced part of the group is Hausdorff and complete in the Z {\displaystyle \mathbb {Z} } adic topology. The group is pure injective, that is, injective with respect to exact sequences where the embedding is as a pure subgroup.Relations with other properties: A torsion-free group is cotorsion if and only if it is algebraically compact. Every injective group is algebraically compact. Ulm factors of cotorsion groups are algebraically compact.
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https://en.wikipedia.org/wiki/Algebraically_compact_group
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In mathematics, in the realm of abstract algebra, a radical polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, if k {\displaystyle k} is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial ∑ i = 1 n x i 2 . {\displaystyle \sum _{i=1}^{n}x_{i}^{2}.} Radical polynomials are characterized as precisely those polynomials that are invariant under the action of the orthogonal group.
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https://en.wikipedia.org/wiki/Radical_polynomial
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The ring of radical polynomials is a graded subalgebra of the ring of all polynomials. The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms, each term being a product of a radical polynomial and a harmonic polynomial. This is equivalent to the statement that the ring of all polynomials is a free module over the ring of radical polynomials. == References ==
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https://en.wikipedia.org/wiki/Radical_polynomial
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In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup of the automorphism group. Some facts: Every inner automorphism is a class automorphism. Every class automorphism is a family automorphism and a quotientable automorphism.
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https://en.wikipedia.org/wiki/Class_automorphism
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Under a quotient map, class automorphisms go to class automorphisms. Every class automorphism is an IA automorphism, that is, it acts as identity on the abelianization. Every class automorphism is a center-fixing automorphism, that is, it fixes all points in the center.
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https://en.wikipedia.org/wiki/Class_automorphism
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Normal subgroups are characterized as subgroups invariant under class automorphisms.For infinite groups, an example of a class automorphism that is not inner is the following: take the finitary symmetric group on countably many elements and consider conjugation by an infinitary permutation. This conjugation defines an outer automorphism on the group of finitary permutations. However, for any specific finitary permutation, we can find a finitary permutation whose conjugation has the same effect as this infinitary permutation.
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https://en.wikipedia.org/wiki/Class_automorphism
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This is essentially because the infinitary permutation takes permutations of finite supports to permutations of finite support. For finite groups, the classical example is a group of order 32 obtained as the semidirect product of the cyclic ring on 8 elements, by its group of units acting via multiplication. Finding a class automorphism in the stability group that is not inner boils down to finding a cocycle for the action that is locally a coboundary but is not a global coboundary.
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https://en.wikipedia.org/wiki/Class_automorphism
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In mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group.
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https://en.wikipedia.org/wiki/SQ-universal_group
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In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theorem and the classification of finite simple groups. Several important infinite groups are CA-groups, such as free groups, Tarski monsters, and some Burnside groups, and the locally finite CA-groups have been classified explicitly. CA-groups are also called commutative-transitive groups (or CT-groups for short) because commutativity is a transitive relation amongst the non-identity elements of a group if and only if the group is a CA-group.
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https://en.wikipedia.org/wiki/CA-group
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In mathematics, in the realm of group theory, a group is said to be capable if it occurs as the inner automorphism group of some group. These groups were first studied by Reinhold Baer, who showed that a finite abelian group is capable if and only if it is a product of cyclic groups of orders n1,...,nk where ni divides ni+1 and nk–1=nk.
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https://en.wikipedia.org/wiki/Capable_group
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In mathematics, in the realm of group theory, a group is said to be critical if it is not in the variety generated by all its proper subquotients, which includes all its subgroups and all its quotients. Any finite monolithic A-group is critical. This result is due to Kovacs and Newman. The variety generated by a finite group has a finite number of nonisomorphic critical groups.
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https://en.wikipedia.org/wiki/Critical_group
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In mathematics, in the realm of group theory, a group is said to be parafree if its quotients by the terms of its lower central series are the same as those of a free group and if it is residually nilpotent (the intersection of the terms of its lower central series is trivial). Parafree groups share many properties with free groups, making it difficult to distinguish between these two types. Gilbert Baumslag was led to the study of parafree groups in attempts to resolve the conjecture that a group of cohomological dimension one is free. One of his fundamental results is that there exist parafree groups that are not free. With Urs Stammbach, he proved there exists a non-free parafree group with every countable subgroup being free.
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https://en.wikipedia.org/wiki/Parafree_group
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In mathematics, in the realm of group theory, a group is said to be superperfect when its first two homology groups are trivial: H1(G, Z) = H2(G, Z) = 0. This is stronger than a perfect group, which is one whose first homology group vanishes. In more classical terms, a superperfect group is one whose abelianization and Schur multiplier both vanish; abelianization equals the first homology, while the Schur multiplier equals the second homology.
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https://en.wikipedia.org/wiki/Superperfect_group
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In mathematics, in the realm of group theory, a group is said to be thin if there is a finite upper bound on the girth of the Cayley graph induced by any finite generating set. The group is called fat if it is not thin. Given any generating set of the group, we can consider a graph whose vertices are elements of the group with two vertices adjacent if their ratio is in the generating set. The graph is connected and vertex transitive.
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https://en.wikipedia.org/wiki/Thin_group_(combinatorial_group_theory)
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Paths in the graph correspond to words in the generators. If the graph has a cycle of a given length, it has a cycle of the same length containing the identity element. Thus, the girth of the graph corresponds to the minimum length of a nontrivial word that reduces to the identity.
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https://en.wikipedia.org/wiki/Thin_group_(combinatorial_group_theory)
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A nontrivial word is a word that, if viewed as a word in the free group, does not reduce to the identity. If the graph has no cycles, its girth is set to be infinity.
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https://en.wikipedia.org/wiki/Thin_group_(combinatorial_group_theory)
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The girth depends on the choice of generating set. A thin group is a group where the girth has an upper bound for all finite generating sets.
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https://en.wikipedia.org/wiki/Thin_group_(combinatorial_group_theory)
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Some facts about thin and fat groups and about girths: Every finite group is thin. Every free group is fat. The girth of a cyclic group equals its order.
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https://en.wikipedia.org/wiki/Thin_group_(combinatorial_group_theory)
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The girth of a noncyclic abelian group is at most 4, because any two elements commute and the commutation relation gives a nontrivial word. The girth of the dihedral group is 2. Every nilpotent group, and more generally, every solvable group, is thin.
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https://en.wikipedia.org/wiki/Thin_group_(combinatorial_group_theory)
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In mathematics, in the realm of group theory, a power automorphism of a group is an automorphism that takes each subgroup of the group to within itself. It is worth noting that the power automorphism of an infinite group may not restrict to an automorphism on each subgroup. For instance, the automorphism on rational numbers that sends each number to its double is a power automorphism even though it does not restrict to an automorphism on each subgroup. Alternatively, power automorphisms are characterized as automorphisms that send each element of the group to some power of that element.
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https://en.wikipedia.org/wiki/Power_automorphism
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This explains the choice of the term power. The power automorphisms of a group form a subgroup of the whole automorphism group. This subgroup is denoted as P o t ( G ) {\displaystyle Pot(G)} where G {\displaystyle G} is the group.
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https://en.wikipedia.org/wiki/Power_automorphism
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A universal power automorphism is a power automorphism where the power to which each element is raised is the same. For instance, each element may go to its cube. Here are some facts about the powering index: The powering index must be relatively prime to the order of each element.
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https://en.wikipedia.org/wiki/Power_automorphism
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In particular, it must be relatively prime to the order of the group, if the group is finite. If the group is abelian, any powering index works. If the powering index 2 or -1 works, then the group is abelian.The group of power automorphisms commutes with the group of inner automorphisms when viewed as subgroups of the automorphism group. Thus, in particular, power automorphisms that are also inner must arise as conjugations by elements in the second group of the upper central series.
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https://en.wikipedia.org/wiki/Power_automorphism
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In mathematics, in the realm of group theory, a quotientable automorphism of a group is an automorphism that takes every normal subgroup to within itself. As a result, it gives a corresponding automorphism for every quotient group. All family automorphisms are quotientable, and particularly, all class automorphisms and power automorphisms are. As well, all inner automorphisms are quotientable, and more generally, any automorphism defined by an algebraic formula is quotientable.
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https://en.wikipedia.org/wiki/Quotientable_automorphism
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In mathematics, in the realm of group theory, a subgroup of a group is said to be centrally closed if the centralizer of any nonidentity element of the subgroup lies inside the subgroup. Some facts about centrally closed subgroups: Every malnormal subgroup is centrally closed. Every Frobenius kernel is centrally closed. SA subgroups are precisely the centrally closed Abelian subgroups. The trivial subgroup and the whole group are centrally closed.
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https://en.wikipedia.org/wiki/Centrally_closed_subgroup
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In mathematics, in the realm of group theory, an IA automorphism of a group is an automorphism that acts as identity on the abelianization. The abelianization of a group is its quotient by its commutator subgroup. An IA automorphism is thus an automorphism that sends each coset of the commutator subgroup to itself. The IA automorphisms of a group form a normal subgroup of the automorphism group. Every inner automorphism is an IA automorphism.
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https://en.wikipedia.org/wiki/IA_automorphism
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In mathematics, in the realm of group theory, the stability group of subnormal series is the group of automorphisms that act as identity on each quotient group.
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https://en.wikipedia.org/wiki/Stability_group
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In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways. In (Hall 1937), a complemented group is one in which every subgroup has a group-theoretic complement. Such groups are called completely factorizable groups in the Russian literature, following (Baeva 1953) and (Černikov 1953). The following are equivalent for any finite group G: G is complemented G is a subgroup of a direct product of groups of square-free order (a special type of Z-group) G is a supersolvable group with elementary abelian Sylow subgroups (a special type of A-group), (Hall 1937, Theorem 1 and 2).Later, in (Zacher 1953), a group is said to be complemented if the lattice of subgroups is a complemented lattice, that is, if for every subgroup H there is a subgroup K such that H ∩ K = 1 and ⟨H, K ⟩ is the whole group.
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https://en.wikipedia.org/wiki/Complemented_group
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Hall's definition required in addition that H and K permute, that is, that HK = { hk: h in H, k in K } form a subgroup. Such groups are also called K-groups in the Italian and lattice theoretic literature, such as (Schmidt 1994, pp.
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https://en.wikipedia.org/wiki/Complemented_group
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114–121, Chapter 3.1). The Frattini subgroup of a K-group is trivial; if a group has a core-free maximal subgroup that is a K-group, then it itself is a K-group; hence subgroups of K-groups need not be K-groups, but quotient groups and direct products of K-groups are K-groups, (Schmidt 1994, pp. 115–116).
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https://en.wikipedia.org/wiki/Complemented_group
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In (Costantini & Zacher 2004) it is shown that every finite simple group is a complemented group. Note that in the classification of finite simple groups, K-group is more used to mean a group whose proper subgroups only have composition factors amongst the known finite simple groups. An example of a group that is not complemented (in either sense) is the cyclic group of order p2, where p is a prime number. This group only has one nontrivial subgroup H, the cyclic group of order p, so there can be no other subgroup L to be the complement of H.
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https://en.wikipedia.org/wiki/Complemented_group
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In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality. There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The only countable Hausdorff Toronto space is the discrete space.The Toronto space problem asks for an uncountable Toronto Hausdorff space that is not discrete. == References ==
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https://en.wikipedia.org/wiki/Toronto_space
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In mathematics, in the realm of topology, a paranormal space (Nyikos 1984) is a topological space in which every countable discrete collection of closed sets has a locally finite open expansion.
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https://en.wikipedia.org/wiki/Paranormal_space
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In mathematics, in the representation theory of algebraic groups, a Grosshans subgroup, named after Frank Grosshans, is an algebraic subgroup of an algebraic group that is an observable subgroup for which the ring of functions on the quotient variety is finitely generated.
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https://en.wikipedia.org/wiki/Grosshans_subgroup
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In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map of algebraic varieties. Finite direct sums and products of rational representations are rational. A rational G {\displaystyle G} module is a module that can be expressed as a sum (not necessarily direct) of rational representations.
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https://en.wikipedia.org/wiki/Rational_representation
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In mathematics, in the representation theory of algebraic groups, an observable subgroup is an algebraic subgroup of a linear algebraic group whose every finite-dimensional rational representation arises as the restriction to the subgroup of a finite-dimensional rational representation of the whole group. An equivalent formulation, in case the base field is closed, is that K is an observable subgroup of G if and only if the quotient variety G/K is a quasi-affine variety. Some basic facts about observable subgroups: Every normal algebraic subgroup of an algebraic group is observable. Every observable subgroup of an observable subgroup is observable.
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https://en.wikipedia.org/wiki/Observable_subgroup
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