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Question: <p>If the Frattini subgroup is trivial,for a group <em>P</em>, then <em>P</em> is elementary abelian. Where, Frattini subgroup is the intersection of all maximal subgroups. Please prove the above statement.</p>
Answer: <p>Let $G$ be a $p$-group. Then every maximal subgroup is normal. </p>
<p>Notice that $G/... | https://math.stackexchange.com/questions/2197158/frattini-subgroup-is-trivial-hen-the-group-is-elementary-abelian |
Question: <p>I have to prove that if $G$ and $G'$ are two finites group of same cardinal, then they are isomorphic.</p>
<p>Actually, it looks obvious. Suppose $G=\{g_1,...,g_n\}$ and $G'=\{h_1,...,h_n\}$. Does the homomorphism $g_i\longmapsto h_i$ work ? </p>
Answer: <p>This is not true in general., For example $\Bb... | https://math.stackexchange.com/questions/1649636/if-g-and-g-are-two-finite-group-of-same-cardinal-then-g-cong-g |
Question: <blockquote>
<p>If <span class="math-container">$G$</span> is perfect, show <span class="math-container">$G/N$</span> is also perfect (for a <span class="math-container">$N\trianglelefteq G$</span>)</p>
</blockquote>
<p>I have some proof, but I don't think it right.</p>
<h3>Proof?</h3>
<p>Choose a <span class... | https://math.stackexchange.com/questions/2573644/if-g-is-perfect-show-g-n-is-also-perfect-for-a-n-trianglelefteq-g |
Question: <p>I'm studying dimension of the group now. But I have some trouble with it.</p>
<p>Consider the dimension of 2 by 2 orthogonal group, <span class="math-container">$O(2)=\{A \in M(2,2,R): AA^T=I\}$</span></p>
<p>Let <span class="math-container">$A = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$</span... | https://math.stackexchange.com/questions/4168724/find-the-dimension-of-the-system-with-equations-and-variables |
Question: <p>this is my first question so let me know if I broke any rules. I am writing my dissertation on the zero divisor conjecture in group rings, and I am struggling to find any examples I could put in my chapter examining the classes of groups mentioned above. This might be a dumb question, but I have searched f... | https://math.stackexchange.com/questions/3544023/looking-for-examples-of-free-groups-ordered-groups-and-unique-product-groups |
Question: <p>Does there exist nontrivial homomorphism between symmetric group $S_n$ to set of integer $\mathbb Z$?</p>
<p>If yes, how many?</p>
<p>If not, then why?</p>
Answer: <p>An element of finite order is by a homomorphism always mapped to an element of finite order. This is proven quite directly from the defin... | https://math.stackexchange.com/questions/2660136/homomorphism-between-s-n-to-mathbb-z |
Question: <p>Let $*$ be defined on $\mathbb R^3$ such that $(x,y,t)*(x',y',t'):=( x+x',y+y',t+t'+\dfrac 12(xy'-x 'y ) )$ , I can show that $(\mathbb R^3 , *)$ is a group ; I want to find the center of this group , Please help . </p>
Answer: <p>Suppose $(x',y',t')$ is in the center. Then $(x,y,t)*(x',y',t')=(x',y',t'... | https://math.stackexchange.com/questions/1474870/let-mathbb-r3-be-a-group-with-operation-x-y-tx-y-t-xx-y |
Question: <p>If you have a group $G$ and a proper subgroup $H$ inside of the group. Then is $H$ a proper subgroup of the quotient group $G/H$? </p>
Answer: <p>In order to be a subgroup of a group, one must first be a subset. However, $H$ is not a subgroup of $G/H$ for any normal subgroup $H$ because $G/H$ contains as ... | https://math.stackexchange.com/questions/676442/quotient-a-group-by-a-proper-subgroup |
Question: <p>I am trying to find an example of a homomorphism from $GL_n(\mathbb{R}) \rightarrow \mathbb{R^+}$ with kernel $K = \{A\in GL_n(\mathbb{R})| \det(A) = \pm 1\}$ however it is eluding me. I can't seem to make it a homomorphism.</p>
Answer: <p>If $\mathbb{R^+}$ is the multiplicative group of positive reals,
t... | https://math.stackexchange.com/questions/1539892/example-of-a-homomorphism-from-gl-n-mathbbr-rightarrow-mathbbr-with |
Question: <p>I am trying to prove (a) - (e) but am struggling with how to start.</p>
<p>$\bf{Question:}$ Let $H,K < G$ and consider the map $f: H \times K \rightarrow G$ given by $f(h,k) = hk$. The image of $f$ we will denote $HK$.</p>
<p>(a) Show that $f$ is injective iff $H \cap K = \{e\}$</p>
<p>(b) Suppose $H... | https://math.stackexchange.com/questions/1482008/understanding-semidirect-products-in-group-theory-through-exercises |
Question: <blockquote>
<p>Prove that the composition of two group homomorphisms is a group homomorphism.</p>
</blockquote>
<p>Let $f:G \longrightarrow G'$ and $g:G' \longrightarrow G''$ be two group homomorphisms.</p>
<p>Let $x$ and $y$ be two arbitrary elements of $G$. Then,</p>
<p>\begin{eqnarray}
(g \circ f)(x ... | https://math.stackexchange.com/questions/2130210/proving-the-composition-of-two-group-homomorphisms-is-a-group-homomorphism |
Question: <p>Let $f:G \rightarrow G'$ be a homomorphism and let $H$ be the kernel of $G$. Suppose $G$ is finite. Show ord$(G)=$ord$(f(G)) \cdot $ord$(H)$.</p>
<p>What I want to do is to construct a bijection, $\Phi$ from $G$\ $H$ (the factor group) to $f(G)$.</p>
<p>This should then tell me, I think, that ord$(G$\ ... | https://math.stackexchange.com/questions/1327197/show-that-the-order-of-g-order-fg-times-order-kerg |
Question: <p>Suppose that G is a finite, nonabelian group with odd order. Show s is surjective, and hence bijective.</p>
<p>I have been told to look at the effects of the squaring map, $s\colon G\to G$, defined by $s(g)=g^2$ on the elements of cyclic groups $\langle g\rangle$ of $G$.</p>
<p>I'm stumped. Could anyone ... | https://math.stackexchange.com/questions/57913/suppose-that-g-is-a-finite-nonabelian-group-with-odd-order-show-s-is-surjectiv |
Question: <p>I know that $|Z_{24}|=24=2^3. 3$, So we can use the equation $t=1+k\,p$ to find the number of sylow groups for each $p=2,3$. Therefore we have $1$ or $4$ sylow $3$-subgroups and $1$ or $3$ sylow $2$-subgroups.</p>
<p>Is $<(12)>$ the Sylow $2$-subgroup and $<(8)>$ the sylow $3$-subgroup?</p>
... | https://math.stackexchange.com/questions/1744801/what-are-the-sylow-subgroups-of-z-24 |
Question: <p>I am working on a question that asks me to list all abelian groups of order $1188$.</p>
<p>I have a list that I will put below that I obtained using the classification of finite abelian groups.</p>
<p>My question is could someone verify whether my answer is correct and is in fact a list of all such group... | https://math.stackexchange.com/questions/2293862/list-of-all-abelian-groups-of-order-1188 |
Question: <p>Are there any groups (not necessarily finite) that have a non-zero even number of elements of order two?</p>
<p>My attempt: since any finite group containing an element of order 2 must be of even order, hence the number of elements of order 2 must be odd, it suffices to find groups of infinite order.</p>
<... | https://math.stackexchange.com/questions/4184824/groups-that-have-a-non-zero-even-number-of-elements-of-order-two |
Question: <p>Let <span class="math-container">$G$</span> be a finitely generated group. For a finite subset <span class="math-container">$S \subset G$</span> which generates <span class="math-container">$G$</span>,</p>
<p>define <span class="math-container">$D_{G,S(n)}$</span> = {<span class="math-container">$s_1 s_2 .... | https://math.stackexchange.com/questions/4189011/compute-limsup-limits-k-rightarrow-infty-l-g-s-k-l-g-t-k |
Question: <p><strong>Do group operators that commute with fixed positive powers necessarily commute with all integer powers as well?</strong></p>
<p>Let's take a group with operators <span class="math-container">$(G, \Omega)$</span> and relax the requirements on each <span class="math-container">$\omega$</span> in <spa... | https://math.stackexchange.com/questions/4193994/groups-with-operators-omega-ni-omega-that-commute-with-positive-powers-doe |
Question: <p>I understand that since $a^6$ belongs to $H$, then $a^{60}=e$. But I am not sure what are the possibilities for the order of $a$?</p>
Answer: <p>We have seen already that the order of <span class="math-container">$a$</span> must divide 60. In fact all divisors of 60 are possible orders of <span class="mat... | https://math.stackexchange.com/questions/2903641/suppose-that-h-is-a-subgroup-of-a-group-g-and-h-10-if-a-belongs-to-g |
Question: <p>Let $E$ be a non empty set. How to prove that there exists $\star:E\times E\rightarrow E$ for which $(E,\star)$ is a group?</p>
Answer: <p>If $E$ is finite, you can just identify $E$ with $\Bbb Z/n\Bbb Z$, where $n=|E|$.</p>
<p>If $E$ is countably infinite, you can similarly identify it with $\Bbb Z$ (or... | https://math.stackexchange.com/questions/2909462/non-empty-set-and-group |
Question: <p>Is this true that <span class="math-container">$N_G(H) \subseteq N_G(H \cap K)$</span> if not what is the counter-example? I am confused in this question?</p>
<p>Clearly <span class="math-container">$g \in N_G(H)$</span> then <span class="math-container">$g(H \cap K)g^{-1}\subseteq H$</span></p>
Answer: ... | https://math.stackexchange.com/questions/2929568/is-this-true-that-n-gh-subseteq-n-gh-cap-k-if-not-what-is-the-counter-ex |
Question: <p><em>Let <span class="math-container">$G$</span> be a finite group and <span class="math-container">$J=\langle g_1,\cdots g_k\rangle$</span> be a sequence of group elements. For any <span class="math-container">$\delta \ge 1$</span>, <span class="math-container">$J$</span> is said to be a cube generating se... | https://math.stackexchange.com/questions/2946994/difficulty-in-understanding-the-definition-of-this-sequence |
Question: <p>I'm trying to see if I can find a bijection between two groups that are infinite of which one in the subset of the other. If I find the inverse <span class="math-container">$\phi^{-1}(x)=\frac{1}{5}x$</span> since it doesn't work for <span class="math-container">$x \in \mathbb{Z}$</span> (because I will ha... | https://math.stackexchange.com/questions/2959686/proving-something-isnt-isomorphic-varphi-5-mathbbz-rightarrow-mathbbz |
Question: <p>Let <span class="math-container">$G_1, G_2$</span> and <span class="math-container">$H$</span> be finite groups such that <span class="math-container">$|G_1|=|G_2|$</span>. Assume that <span class="math-container">$|L(G_1)|<|L(G_2)|$</span>. Is there a way to show that <span class="math-container">$|L(G... | https://math.stackexchange.com/questions/2968181/inequality-involving-the-number-of-subgroups-of-a-direct-product-of-groups |
Question: <p>Do we just use the subgroup criteria on this? Finding that it's closed, has inverse and identity within the subgroup?</p>
<p>But I still don't how does that prove the fact that's <span class="math-container">$H = \{2^n:𝑛 \in\mathbb{Z}\}$</span> is a subset of <span class="math-container">$\mathbb{Q}\setm... | https://math.stackexchange.com/questions/2993916/show-that-the-set-h-2n-in-mathbbz-is-a-subgroup-of-for-mathbbq |
Question: <p>Past year exam question:</p>
<p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$H$</span> a subgroup of <span class="math-container">$G$</span>. Suppose there exist some <span class="math-container">$g \in G$</span>, <span class="math-container">$g \notin H$</span>... | https://math.stackexchange.com/questions/2999699/let-h-a-subgroup-of-g-exists-g-in-g-g-notin-h-such-that-gh-hg-a |
Question: <p>Let <span class="math-container">$G$</span> be a finite group and <span class="math-container">$K$</span> a normal subgroup of <span class="math-container">$G$</span>. Suppose <span class="math-container">$|K|^2 \nmid |G|$</span> and <span class="math-container">$K$</span> is simple, Prove <span class="mat... | https://math.stackexchange.com/questions/3004107/let-g-be-a-finite-group-and-k-a-normal-subgroup-of-g-suppose-k2-nmid |
Question: <p>“Prove that the additive group of real numbers doesn’t have any proper subgroup with finite index.”
I want to know how to prove this.</p>
Answer: <p>Assume that <span class="math-container">$G\leq\mathbb{R}$</span> is a subgroup with <span class="math-container">$[\mathbb{R}:G]=n\in\mathbb{Z}_{>0}$</sp... | https://math.stackexchange.com/questions/3005921/proof-of-group-theory |
Question: <p>Consider a group and its subgroup:
<span class="math-container">$$\,K\,<\,G\,\;.$$</span>
An element <span class="math-container">$g\in G$</span> generates a bijection of <span class="math-container">$\,G\,$</span> onto itself,
<span class="math-container">\begin{eqnarray}
\hat{\cal{L}}_g\;\colon\qu... | https://math.stackexchange.com/questions/3008670/a-simple-question-on-groups-and-quotient-spaces |
Question: <p>Is there a compact matrix group <span class="math-container">$G\subseteq GL(n,\mathbb{R})$</span> such that <span class="math-container">$|G|$</span> is countable infinite?</p>
Answer: <p>So the answer is negative. In general there is no countably infinite and compact Hausdorff group. See this:</p>
<p><a... | https://math.stackexchange.com/questions/3023788/is-there-such-a-group |
Question: <blockquote>
<p>Suppose centraliser of group of element has order 4 . Then what information can we deduct about center </p>
</blockquote>
<p>I know that <span class="math-container">$Z(G)\subset Z(x)$</span> where <span class="math-container">$|Z(x)|=4$</span></p>
<p>Any group of order 4 is abelian <span ... | https://math.stackexchange.com/questions/3028409/how-to-deduce-information-about-center-of-group-if-only-order-of-centraliser-is |
Question: <blockquote>
<p>Let <span class="math-container">$G$</span> a group and let <span class="math-container">$A,B$</span> subgroups of <span class="math-container">$G$</span>. Let a map <span class="math-container">$f:A/A\cap B\to G/B$</span> by <span class="math-container">$(A\cap B)a\mapsto Ba$</span>. Prove ... | https://math.stackexchange.com/questions/3077227/prove-that-a-map-is-defined-properly |
Question: <p><span class="math-container">$G$</span> is a Group of order <span class="math-container">$pq$</span>, if <span class="math-container">$G$</span> has exactly one subgroup of order <span class="math-container">$p$</span> and another with order <span class="math-container">$q$</span>, then <span class="math-c... | https://math.stackexchange.com/questions/3141832/g-is-a-group-of-order-pq-if-g-has-exactly-one-subgroup-of-order-p-and-a |
Question: <p>Let <span class="math-container">$M$</span> be a non prime number and <span class="math-container">$G$</span> be the set of non-zero integers modulo <span class="math-container">$M$</span>, under multiplication modulo <span class="math-container">$M$</span>.
Show this is not a group.</p>
<p>My attempt:
Si... | https://math.stackexchange.com/questions/3145241/modulo-groups-and-non-prime-numbers |
Question: <p>It's easy to show that <span class="math-container">$\mathbb{Q}$</span> and <span class="math-container">$\mathbb{Q}\times\mathbb{Q}$</span> are not ring isomorphic as the first one has no zero divisors where as the second one has zero divisors. But I can't find any solution in case of group isomorphism.</... | https://math.stackexchange.com/questions/3151523/how-do-i-show-that-mathbbq-and-mathbbq-times-mathbbq-are-not-group-i |
Question: <p>I know that if <span class="math-container">$z\in Z(G)$</span>, the centre of group <span class="math-container">$G$</span> then it is true that <span class="math-container">$cl(z)=\{z\}$</span> where <span class="math-container">$cl(g)$</span> is the conjugacy class that contains element <span class="mat... | https://math.stackexchange.com/questions/3151972/if-clg-g-can-we-imply-that-g-in-zg |
Question: <blockquote>
<p><span class="math-container">$\phi \in \operatorname{Aut}(\Bbb Z_{50})$</span> via <span class="math-container">$\phi(11) = 3$</span> Then <span class="math-container">$\phi(x) = $</span>? For any <span class="math-container">$x \in \Bbb Z_{50}$</span></p>
</blockquote>
<p>The answer is <sp... | https://math.stackexchange.com/questions/3153385/phi-in-operatornameaut-bbb-z-50-via-phi11-3-then-phix |
Question: <blockquote>
<p>Let <span class="math-container">$G$</span> be a group. <span class="math-container">$|G|=21$</span> and <span class="math-container">$|Z(G)| \neq 1$</span> <span class="math-container">$\Rightarrow$</span> <span class="math-container">$|Z(G)| = $</span> ?</p>
</blockquote>
<p>We know that ... | https://math.stackexchange.com/questions/3153429/let-g-be-a-group-g-21-and-zg-neq-1-rightarrow-zg |
Question: <blockquote>
<p>Let <span class="math-container">$G$</span> be a group with order 12. Which of the following claims are false?</p>
</blockquote>
<p>Does Lagrange's Theorem imply that there could exist a subgroup with order 6? I'm not sure where to begin. The question was taken from a past test with multipl... | https://math.stackexchange.com/questions/3153581/let-g-be-a-group-with-order-12-which-of-the-following-claims-are-false |
Question: <p>Let
<span class="math-container">\begin{align*}
G=\langle s, t \mid s^4=1, s^2 = t^3 \rangle
\end{align*}</span>
and
<span class="math-container">\begin{align*}
G'=\langle S, T \mid S^2=1, T^3=1 \rangle
\end{align*}</span>
be two groups. Are these groups isomorphic to each other? Thank you very much.</p>
... | https://math.stackexchange.com/questions/3161957/are-the-two-groups-g-g-isomorphic |
Question: <blockquote>
<p><span class="math-container">$H$</span> is a subgroup of <span class="math-container">$G$</span>, prove <span class="math-container">$|gHg^{-1}| = |H|$</span>, <span class="math-container">$\forall g \in G$</span></p>
</blockquote>
<p>Here's what I know:</p>
<p>If <span class="math-contain... | https://math.stackexchange.com/questions/3194039/h-is-a-subgroup-of-g-prove-ghg-1-h-forall-g-in-g |
Question: <blockquote>
<p><span class="math-container">$H$</span> is a subgroup of <span class="math-container">$G$</span>, <span class="math-container">$N$</span> is a normal subgroup of <span class="math-container">$G$</span>, <span class="math-container">$\gcd(|H|, |G/N|) = 1 \Rightarrow$</span> <span class="math-... | https://math.stackexchange.com/questions/3195123/h-is-a-subgroup-of-g-n-is-a-normal-subgroup-of-g-gcdh-g-n-1 |
Question: <p>I want to use this and the fact that <span class="math-container">$H_n \cap H_m = \{1\}$</span> which I've already proved to show that <span class="math-container">$G \cong H_n \times H_m$</span></p>
<p>Since this are subgroups, and are the only subgroups like this, <span class="math-container">$H_n$</spa... | https://math.stackexchange.com/questions/3221041/is-h-nh-m-g-if-g-nm-and-h-n-h-m-are-the-only-subgroups-of-g-of-orde |
Question: <p>Let <span class="math-container">$H$</span> be an infinite cyclic subgroup of a group <span class="math-container">$G$</span>. </p>
<p>Is the quotient group <span class="math-container">$G/C_G (H)$</span> of order two?</p>
Answer: <p>As pointed out in the comments, this is not true. Any abelian group i... | https://math.stackexchange.com/questions/3222863/about-centralizer-of-an-infinite-cyclic-subgroup |
Question: <p>I am not sure exactly how to phrase this problem so I appologise if it is not clear, also this is somewhat long but I wanted to explain exactly where I was with the problem. If you have any questions feel free to ask.</p>
<hr>
<p><strong>Description of Problem</strong></p>
<p>Given a set of variables <s... | https://math.stackexchange.com/questions/3240948/are-there-equations-which-have-solutions-in-all-groups-but-which-are-not-algebra |
Question: <p>I want to find an example of a homomorphism <span class="math-container">$f : \mathbb{Z}^2 \ast \mathbb{Z}^2 \to \mathbb{Z}^2$</span> such that <span class="math-container">$\ker f$</span> is free and not finitely generated.</p>
<p>My idea is to define <span class="math-container">$f$</span> on each copy ... | https://math.stackexchange.com/questions/3241462/proving-that-there-is-a-homomorphism-f-mathbbz2-ast-mathbbz2-to-ma |
Question: <p>Wikipedia has a basic <a href="https://en.wikipedia.org/wiki/*-algebra#*-ring" rel="nofollow noreferrer">reference on *-rings in the *-algebra article</a>, which defines a *-ring as a ring equipped with a <code>*</code> operator which is an antiautomorphism and an involution.</p>
<p>However, I am not inte... | https://math.stackexchange.com/questions/3259366/groups-or-group-with-involution |
Question: <p>In this <a href="https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)" rel="nofollow noreferrer">page</a> of wikipedia there is a disproving of the converse of Lagrange's theorem. I would like to see a more simple (or short) disproving of Lagrange's theorem.</p>
Answer: <p>I prefer the follow... | https://math.stackexchange.com/questions/3266823/disproving-the-converse-of-lagranges-theorem |
Question: <p>Suppose that <span class="math-container">$(A,+)$</span> is an abelian group and that <span class="math-container">$A=B \cup C$</span>. Define for any <span class="math-container">$X \subseteq A$</span> the following</p>
<p><span class="math-container">$$\Delta X = \{ x_1-x_2 ; x_1 , x_2 \in X \}$$</span... | https://math.stackexchange.com/questions/3274585/prove-that-either-a-delta-b-or-a-delta-c |
Question: <p>Let <span class="math-container">$(G,*)$</span> be a group. And let <span class="math-container">$ x$</span> be a element of odd order of <span class="math-container">$G$</span> , then prove or disprove that , there is a element <span class="math-container">$y$</span> in <span class="math-container">$G$</s... | https://math.stackexchange.com/questions/3278714/prove-or-disprove-that-there-is-a-element-y-in-g-such-that-y2-x |
Question: <p>I am trying to learn group theory from the textbook Algebra by Michael Artin 2nd edition. I am looking for finite group and subgroup examples that are useful for developing an intuition about group theory. So far, I have been using the symmetric group <span class="math-container">$S_3$</span>. </p>
<p>Pre... | https://math.stackexchange.com/questions/3289459/what-examples-to-use-when-learning-group-theory |
Question: <p>Let <span class="math-container">$G$</span> be the smallest group of <span class="math-container">$2 \times 2$</span> matrices whose entries are complex numbers which contains both matrices<br>
<span class="math-container">$$\begin{bmatrix}0&1\\-1&0\end{bmatrix} \quad \text{ and } \quad \begin{bmat... | https://math.stackexchange.com/questions/3304910/group-of-2-times-2-matrices-with-complex-numbers |
Question: <p>Let <span class="math-container">$H$</span> be a subgroup of finite index in group <span class="math-container">$G$</span> and let <span class="math-container">$g \in G$</span>. </p>
<p><strong>Question:</strong> Is it true, that <span class="math-container">$C_H(g)$</span> has finite index in <span class... | https://math.stackexchange.com/questions/3315817/finite-index-of-centralizer-c-hg-in-c-gg |
Question: <p>If <span class="math-container">$H$</span> and <span class="math-container">$K$</span> are subgroups of a group <span class="math-container">$G,*$</span>, how can one proof that:
<span class="math-container">$$HK = \operatorname{grp}\{H \cup K\} \iff HK \text{ is a subgroup of } G$$</span> </p>
<p>I've se... | https://math.stackexchange.com/questions/3321901/proof-for-hk-operatornamegrp-h-cup-k-iff-hk-text-is-a-subgroup-of |
Question: <p>Let G be a group. If order of the element in the group <span class="math-container">$G$</span> is prime (<span class="math-container">$G$</span> is not a <span class="math-container">$p$</span>-group, order of the elements are different prime). Is it true that every element of order <span class="math-conta... | https://math.stackexchange.com/questions/3329586/let-g-be-a-group-if-order-of-the-element-in-the-group-g-is-prime |
Question: <blockquote>
<p>Let <span class="math-container">$G$</span> be group, <span class="math-container">$S \subset G$</span>, <span class="math-container">$g \in G $</span>. Prove:
<span class="math-container">$$gC_{G}(S)(g)^{-1} = C_{G}(gS(g)^{-1}), \forall g\in G.$$</span></p>
</blockquote>
<p>I have manage... | https://math.stackexchange.com/questions/3333011/let-g-be-group-s-subset-g-g-in-g-prove-gc-gsg-1-c-gg |
Question: <blockquote>
<p>If we know <span class="math-container">$\mathbb{Z}_{n}/ \mathbb{Z}_{m} \cong \mathbb{Z}_{k}$</span>, can we conclude that <span class="math-container">$\mathbb{Z}_{n} \cong \mathbb{Z}_{m} \times \mathbb{Z}_{k} $</span>?</p>
</blockquote>
<p>I think not, but I can not find right counterexam... | https://math.stackexchange.com/questions/3333937/if-we-know-mathbbz-n-mathbbz-m-cong-mathbbz-k-can-we-conclud |
Question: <p>Let <span class="math-container">$S_5$</span> denote the group of bijections of the {1, 2, 3, 4, 5} under composition. Find all the elements <span class="math-container">$f ∈ S_5$</span> of order two such that <span class="math-container">$f(1) = 2$</span></p>
<p>I know for order of 2 we need that <span c... | https://math.stackexchange.com/questions/3358628/find-all-elements-f-%e2%88%88-s-5-of-order-2-such-that-f1-2 |
Question: <p>Given groups <span class="math-container">$H$</span> and <span class="math-container">$K$</span>, define a group <span class="math-container">$G$</span> based on <span class="math-container">$H$</span> and <span class="math-container">$K$</span> as <span class="math-container">$G:=\{(h,k):h\in H, k\in K\}$... | https://math.stackexchange.com/questions/3369151/define-group-g-based-on-h-and-k |
Question: <blockquote>
<p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$a,b\in G$</span>. Prove that <span class="math-container">$|a|=|a^{-1}|$</span> and that <span class="math-container">$|ab|=|ba|$</span>.</p>
</blockquote>
<p>I said that <span class="math-container">$|... | https://math.stackexchange.com/questions/3383518/let-g-be-a-group-and-a-b-in-g-prove-that-a-a-1-and-that-ab-ba |
Question: <p>I know the order say m is some integral integer that gives gives an element an identity, say 0 in the additive case. Can somebody give an example as to why it has infinite order? Is that because there is infinity amount of elements?</p>
Answer: <p>Name <span class="math-container">$G$</span> one of those ... | https://math.stackexchange.com/questions/3388664/in-the-additive-groups-mathbbz-mathbbq-mathbbr-or-mathbbc |
Question: <p>I am working my way through Charles Pinter's book: A Book of Abstract Algebra. From recommendations on this site, I found a page/web address on Wisconsin University's Math Department that provides solutions to many (perhaps all) of the abundant exercises that are present in Pinter's book. </p>
<p>One of t... | https://math.stackexchange.com/questions/3397903/true-or-false-if-the-product-of-n-elements-of-a-group-is-the-identity-element |
Question: <p>Explicitly construct the regular representation of <span class="math-container">$\mathbb Z_3$</span> and diagonalize it. Since we are now fully diagonal every entry must furnish a one dimensional irreducible representation. Do you think you recovered all the irreps we have found in class? </p>
Answer: <p>... | https://math.stackexchange.com/questions/3412322/construct-the-regular-representation-of-z3-and-diagonalize-it |
Question: <p>Does converse of Lagrange's theorem hold in <span class="math-container">$A_{4} \times \Bbb Z_{2}?$</span></p>
<p>The order of this group is <span class="math-container">$24$</span> and I'm unable to find a subgroup of order 4.Does there exist any group of order <span class="math-container">$4$</span> in ... | https://math.stackexchange.com/questions/3436892/does-converse-of-lagranges-theorem-hold-in-a-4-times-bbb-z-2 |
Question: <p>Suppose I am trying to construct a semidirect product and I have two homomorphisms <span class="math-container">$\varphi_1:K\rightarrow\text{Aut}(H)$</span> and <span class="math-container">$\varphi_2:K\rightarrow\text{Aut}(H)$</span>. Furthermore, suppose that <span class="math-container">$\ker\varphi_1\c... | https://math.stackexchange.com/questions/3440119/homomorphisms-with-the-same-kernel-has-the-same-semidirect-product |
Question: <blockquote>
<p>Find all subgroups of <span class="math-container">$(\Bbb{Z}_2\times\Bbb{Z}_4,\overline{+})$</span>.</p>
</blockquote>
<p>I could find the following subgroups:</p>
<p><span class="math-container">$$\begin{array}{ll}
H_1=\langle(0,0)\rangle=\{(0,0)\}&\text{(Trivial subgroup)}\\
H_2=\lan... | https://math.stackexchange.com/questions/3441019/find-all-subgroups-of-bbbz-2-times-bbbz-4 |
Question: <p>Question is that we have an element <span class="math-container">$a$</span>, where <span class="math-container">$a = p-1$</span> and <span class="math-container">$p$</span> is a prime number, and I need to prove that order of <span class="math-container">$a$</span> is <span class="math-container">$2$</span... | https://math.stackexchange.com/questions/3445321/how-to-show-that-order-of-an-element-is-2-where-the-element-is-p-1-for-prim |
Question: <p>Let <span class="math-container">$G$</span> be a finite group of order <span class="math-container">$n$</span> given by generator-relator representation. </p>
<p><strong>Question :</strong> Let <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are two elements of <span cl... | https://math.stackexchange.com/questions/3460030/group-operation-in-generator-relator-representation |
Question: <p>This is part of an assignment, so please no full answers just hints (c:</p>
<p>Let <span class="math-container">$a=(1234)$</span>, <span class="math-container">$b=(13)(5678)$</span>, <span class="math-container">$G=\langle a,b\rangle$</span>.
Show that the quotient group <span class="math-container">$G/\l... | https://math.stackexchange.com/questions/3485924/show-that-a-quotient-group-is-cyclic |
Question: <p>Let T be the set of "twists" that a curve can express. A twist of a curve is essentially a normal vector to the curve parameterized by the curve. The normal vector may include angle in 3D.</p>
<p>In a mobius strip there is a twist vector that simply is the normal vector to the strip. In 2D it would be sim... | https://math.stackexchange.com/questions/3514356/the-twist-group |
Question: <p>We defined a Folner sequence by the condition that <span class="math-container">$$ \lim_{n \to \infty} \frac{ | g A_n \Delta A_n |}{ | A_n|} = 0 $$</span> for all <span class="math-container">$g \in G$</span>, where <span class="math-container">$\left\{A_n \right\}$</span> is a sequence of finite subsets o... | https://math.stackexchange.com/questions/3526927/how-to-show-this-is-a-folner-sequence |
Question: <p>Let U be the submodule of the group algebra <span class="math-container">$ℂ[S_3]$</span> generated by</p>
<p><span class="math-container">$u_1 := 1−(2 1) + (3 2)−(3 1 2)$</span></p>
<p>For all <span class="math-container">$τ ∈ S_3$</span>, let <span class="math-container">$U_τ$</span>
:= {<span class="ma... | https://math.stackexchange.com/questions/3528930/prove-that-u-%e2%8a%95u-2-1-u-%e2%8a%95u-3-1 |
Question: <p>It's easy to construct a countable series of distinct groups - the cyclic groups, for instance - and it's also easy to create a family of groups parametrized by the reals, but most such constructions will have isomorphisms betwteen most if not all of the groups.</p>
<p>As the category Group is very large,... | https://math.stackexchange.com/questions/3546089/is-there-a-natural-family-of-nonisomorphic-groups-parametrized-by-mathbbr |
Question: <p>On page 40 of the first edition of Artin, he writes: </p>
<blockquote>
<p>Going back to a general law of composition, suppose we want to define a product of a string of <span class="math-container">$n$</span> elements of a set:
<span class="math-container">$$a_1 a_2 \ldots a_n = ?$$</span>
There ar... | https://math.stackexchange.com/questions/3550592/artin-on-the-associative-law-of-n-elements |
Question: <p>I am studying about subgroup. My definition of subgroup is that:</p>
<p>Let a set <span class="math-container">$G$</span>, with a binary operation<span class="math-container">$
×:G×G→G,(a,b)↦×(a,b)=:a×b$</span> be a group. Then <span class="math-container">$H⊂G$</span> is a subgroup iff <span class="math-... | https://math.stackexchange.com/questions/3591734/why-is-it-true-that-if-h-is-a-subgroup-of-a-group-g-then-1-h-1-g |
Question: <p>Øystein Ore in 'Some remarks on commutators' proof that:</p>
<p>''any element of the alternating group <span class="math-container">$A_n$</span> with n ≥ 5 is a commutator of two elements''.</p>
<p>I quest the case <span class="math-container">$n=4$</span>. I reasoned that:</p>
<p>1) <span class="math-c... | https://math.stackexchange.com/questions/3591908/commutator-element-in-permutation-group |
Question: <p>As order of <span class="math-container">$Aut(G)$</span> is prime no. Then this implies <span class="math-container">$Aut(G)$</span> is cyclic this means <span class="math-container">$Aut(G)$</span> is abelian this implies inner automorphism group is also cyclic, as cyclic subgroup of cyclic group is cycli... | https://math.stackexchange.com/questions/3618620/let-g-be-finite-group-and-order-of-autg-is-prime-number-p-then-prove-tha |
Question: <p>In my book, "Elements of Modern Algebra, 7th ed.- /Gilberts"</p>
<p>Characterization of a Subring is given as following (that is, conditions to be a subring of a ring <span class="math-container">$R$</span>)</p>
<blockquote>
<p>A subset <span class="math-container">$S$</span> of a ring <span clas... | https://math.stackexchange.com/questions/2768948/characterization-of-a-subring-shouldnot-we-concern-ab-and-ba-simultaneously |
Question: <p>Let $G$ be a group, $N\triangleleft G$ an index-two (normal) subgroup, and $H_1,H_2<G$ two subgroups.
Is it true that
$$H_1\cap N = H_2\cap N \Rightarrow H_1 = H_2\ ?$$
If no, is it true with the extra hypothesis that $H_2=gH_1g^{-1}$ for some $g\in G$?</p>
<p>Proofs or couterexamples would be apprecia... | https://math.stackexchange.com/questions/2735423/index-two-subgroup-implies-full-group |
Question: <p>What is the order of element 2 in group ($\Bbb{Z^*_{47}}$,x) ?</p>
<p>Got no clue from where to start any help would be great</p>
Answer: <p>Let us use that computer help. This and similar problems can be best covered by using computer algebra systems, like <a href="http://www.mathsage.org" rel="nofollow... | https://math.stackexchange.com/questions/2763690/order-of-element-2-in-group-bbbz-47-under-multiplication |
Question: <blockquote>
<p>Let $\langle G,\ast \rangle$ be a group and $X$ be a set. We define $F$ to be the set of all the functions from $X$ to $G$, meaning $F=\{ f\mid f:X \rightarrow G\}$.<br>
We define the operation $\bullet$ on the elements of $F$ by the following:
$$\forall f,g \in F, \forall x \in X: (f \b... | https://math.stackexchange.com/questions/2807293/prove-that-langle-f-bullet-rangle-is-a-group-where-f-f-mid-fx%e2%86%92g-and |
Question: <p>Question: Consider the following presentation $G=\langle a,b|aba^{-1}b^{-1} \rangle$. Does there exist a Van Kampen diagram over the presentation whose boundary label is the word $w=a^{2}ba^{-1}b^{-2}a^{-1}b$?</p>
<p>What I tried: First, the relators are $aba^{-1}b^{-1}$, $bab^{-1}a^{-1}$ and all their cy... | https://math.stackexchange.com/questions/2808012/van-kampen-diagram-van-kampen-lemma |
Question: <p>Apparently $\langle t_L,t_{L/2}r_h, r_v \rangle $ is equal to $ \langle t_L,t_{L/2}r_h,r_hr_v \rangle $ where each of these groups is described by a set of generators with $t_L$ representing a translation by a distance L, $t_{L/2}r_h$ is a glide reflection, $r_v$ is vertical reflection and $r_hr_v$ is a ... | https://math.stackexchange.com/questions/2764504/equal-generator-sets |
Question: <p>I have some issues visualizing how to compute quotient of subgroups in general, safe making explicit the cosets which is not really satisfactory nor geometrically meaningful.</p>
<p><strong>Here is a specific instance.</strong> Let $k$ and $K$ two fields, $K$ extending $k$. This could also be rings, like ... | https://math.stackexchange.com/questions/2754340/on-quotients-of-unipotent-subgroups |
Question: <blockquote>
<p>Let G = $<a>$ be a cyclic group of order n. Prove that the cyclic subgroup generated by $a^m$ is the same as the cyclic subgroup generated by $a^d$, where d = (m, n)</p>
</blockquote>
<p>The book said it suffices to show that $a^d$ is a power of $a^m$. <br>
I proved it using d = mu + ... | https://math.stackexchange.com/questions/2809732/cyclic-subgroup-generated-by-am-is-the-same-as-the-cyclic-subgroup-generated |
Question: <p>Let $K$ be an algebraically closed field of characteristic $p>0$ and let
$n$ be a power $p$. In this case, the abstract groups $SL_{n}(K)$ and $PGL_{n}(K)$ are isomorphic under the natural map $\alpha:SL_{n}(K)\to PGL_{n}(K)$.</p>
<p><strong>Question 1.</strong> Are they also isomorphic as algebraic g... | https://math.stackexchange.com/questions/2810059/isomorphy-of-abstract-vs-algebraic-groups |
Question: <p>Let $G$ be a group and let $g\in G$. Prove that if $g\neq 1$, then there exists a subgroup of $G$ which is maximal with respect to the property of not containing $g$.</p>
<p>I built a chain of subgroups $M_\lambda$ for $\lambda\in \Lambda$ (where each subgroup not contain $g$) partially ordered by inclusi... | https://math.stackexchange.com/questions/2810304/existence-of-maximal-subgroup-of-g-with-the-property-of-not-containing-g-in-g |
Question: <p>In the proof that the multiplication defined on HK (H and K being groups) is associative (Dummit and Foote p. 176) I do not understand the following step (the dot operation is the action of H on K, a, b and c are elements of H and x, y and z are elements of K):
(a x.b x.(y.c), xyz) = (a x.(b y.c), xyz)</p>... | https://math.stackexchange.com/questions/2791868/group-action-on-another-group-preserves-the-latters-group-operation |
Question: <p>Is finiteness a Quasi-Isometric invariant property?!
i.e.
Let $G$,$H$ be two groups which $G$ is finite and $G\sim_{QI} H$, is $H$ finite?!</p>
Answer: <p>A f.g. group $G$ is quasi-isometric to the trivial group iff every $g\in G$ has bounded length (with respect to some generating set), i.e., iff $G$ is ... | https://math.stackexchange.com/questions/2825097/quasi-isometry-and-finiteness |
Question: <p>Let $n=2^7 \cdot 3^5 \cdot 11^3 \cdot 35$. In how many ways can the cyclic group $C_n$ can be written as a direct product of two or more nontrivial groups? List all these direct products.</p>
<p>Can someone guide me how to do this question please. I am not looking for a straight answer obviously. </p>
<p... | https://math.stackexchange.com/questions/1502639/direct-product-of-nontrivial-groups |
Question: <p>Continuing for my study on practicing group theory...
I am now stuck on this problem about composition factors,</p>
<blockquote>
<p>$G$ is a group such that $|G|=p^2q$ where $p \neq q$ and $p,q$ are prime. Prove that the composition factors of $G$ are $C_p,C_p,C_q$ in some order.</p>
</blockquote>
<p>W... | https://math.stackexchange.com/questions/1555306/application-of-burnsides-theorem-composition-factors-of-g-p2q-are |
Question: <p>I'm trying to prove that if $G$ is an Abelian group under $\cdot$, $\forall a,b \in G. \forall z \in \mathbb{Z}. (a \cdot b)^n = a^n \cdot b^n.$ I was originally considering doing this problem using an AFSOC, but I realized that originally assuming that $(a \cdot b)^n \neq a^n \cdot b^n$ would be rather di... | https://math.stackexchange.com/questions/1555799/properties-of-abelian-groups |
Question: <p>I've been starting to play around with some properties of groups, and I wanted to prove this claim to make things simpler for me in the future. If $G$ is closed under and operation $\cdot$ that is also associative, I want to show that $\forall a_1, a_2, ..., a_n \in G$, no matter how I choose to bracket $a... | https://math.stackexchange.com/questions/1555868/induction-with-an-associative-operator |
Question: <p>I have a problem on the notion of FCelement.</p>
<p>let $G$ be a group and $a\in G$ of finite order, so $\langle a \rangle$ is an FC group.</p>
<p>My qeustion is : Why $a$ could be a non FC element in $G$? </p>
Answer: <p>Consider the group of all permutations of natural numbers, $G=S_\mathbb{N}$. Let $... | https://math.stackexchange.com/questions/1590843/about-the-concept-of-an-fc-element |
Question: <p>Good day all,</p>
<p>Wikipedia states:
There are 2328 groups of order 128 up to ismorphism. How is this calculated? Also, what does "up to isomorphism" mean? </p>
<p>(I know what an isomorphism is...i'm just unsure of the phrasing "up to...")
Thanks!</p>
Answer: <p>The groups $(\Bbb Z/5\Bbb Z)^\times$ a... | https://math.stackexchange.com/questions/1534592/counting-the-number-of-abelian-groups-of-a-certain-order |
Question: <p>How do I prove that any finite subgroup of $SO(2)$ must be cyclic? </p>
<p>Also, what are all the finite subgroups of $O(2)$?</p>
Answer: <p>I am not quite sure what you mean by "prove" in this case, but $SO(2)$ is the group of rotations in the plane. A finite subgroup must be the group of rotations by a... | https://math.stackexchange.com/questions/1571081/proof-that-any-finite-subgroup-of-so2-must-be-cyclic |
Question: <p>This may be a stupid question, but </p>
<p>let's consider the cyclic group $G=(\mathbb{Z}/10\mathbb{Z},+)=\{0,1,2,3,4,5,6,7,8,9\}$.</p>
<p>By Lagrange's Theorem this group can only have subgroups of order $2$ or $5$, since its order is $10$.</p>
<p>So for example the group $H=(\mathbb{Z}/3\mathbb{Z},+)=... | https://math.stackexchange.com/questions/1535672/why-is-h-not-a-subgroup-of-g |
Question: <p>An amenable group is at most contable group $G$ for which exist a sequence $\{F_n\}_{n\in \mathbb{N}}$ of finite sets such that
$$\displaystyle\lim_{n\rightarrow \infty}\frac{|(g\cdot F_n)\triangle F_n|}{|F_n|}=0 $$
for every $g\in G$.</p>
<p>There is something that I am not understanding about this defin... | https://math.stackexchange.com/questions/1593043/about-the-definition-of-amenable-group |
Question: <blockquote>
<p>Show that $D_{2n}$ has two conjugacy classes if $n$ is even, but only one if $n$ is odd.</p>
</blockquote>
<p>My questions:</p>
<ol>
<li><p>What is meant by 'two conjugacy classes of reflections'</p></li>
<li><p>How is this question related to whether $n$ is even or odd</p></li>
</ol>
<p>... | https://math.stackexchange.com/questions/1559139/show-that-d-2n-has-two-conjugacy-classes-of-reflections-if-n-is-even-but |
Question: <p>Calculate how groups up isomorphisms exist of order 88 such that has least one element of order 8.</p>
<p>I have two groups: $\mathbb{Z}_{88}$ (abelian) and $\mathbb{Z_{11}}\rtimes_{\phi}\mathbb{Z}_8$, where $\phi:\mathbb{Z}_{8}\rightarrow Aut(\mathbb{Z}_{11})$, defined by $y\mapsto \phi(y)(a)=a^{10}$, $a... | https://math.stackexchange.com/questions/1572728/how-many-groups-up-isomorphisms-exist-such-that-has-least-one-element-of-order-8 |
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