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Corollary 7.9. Assume that \( H \) is contained in the center of \( G \) and that \( \left| H\right| \) and \( \left| {G/H}\right| \) are relatively prime. Then \( G \simeq H \times G/H \) . | Proof. This follows from cor.7.8, since \( N \) centralizes \( H \) . | No |
Theorem 7.10. Assume that \( H \) is abelian and normal in \( G \) . Let \( g \in G \) ; let \( f \) be the smallest integer \( > 0 \) such that \( {g}^{f} \in H \) ; let \( {C}_{g} \) be the cyclic subgroup generated by \( g \) . Then :\n\n\[ \n\operatorname{Ver}\left( g\right) = \mathop{\prod }\limits_{{\gamma \in G/... | Proof. Let \( {\gamma }_{\alpha } \) be a set of representatives of \( G/{C}_{g}H \), and let \( {x}_{\alpha } \) be the image of \( {\gamma }_{\alpha }^{-1} \) in \( X = G/H \) . The orbits of \( {C}_{g} \) in \( X \) are the \( {C}_{g}{x}_{\alpha } \) . We may then apply formula (7.3) with \( {z}_{\alpha } = {\gamma ... | Yes |
Corollary 7.11. If \( h \in H \), then \( \operatorname{Ver}\left( h\right) = \mathop{\prod }\limits_{{\gamma \in G/H}}{\gamma h}{\gamma }^{-1} \) . | Proof. This is the special case \( f = 1 \) . | No |
Proposition 7.12. If \( G \) is a torsion-free group with a finite index subgroup \( H \) isomorphic to \( \mathbf{Z} \), then \( G \) is isomorphic to \( \mathbf{Z} \) . | Proof. After replacing \( H \) by the intersection of its conjugates, we may assume that \( H \) is normal in \( G \) . The group \( G \) acts on \( H \) by conjugation and this action yields a group homomorphism \( \varepsilon : G \rightarrow \operatorname{Aut}\left( H\right) = \{ \pm 1\} \) . There are two cases:\n\n... | Yes |
Theorem 7.13. Let \( H \) be a p-Sylow subgroup of \( G \) and let \( \varphi : H \rightarrow A \) be a homomorphism with values in a finite abelian p-group A. Then :\n\n(1) \( \varphi \) extends to a homomorphism of \( G \) into \( A \) if and only if, for all \( h,{h}^{\prime } \in H \) conjugate in \( G \), we have ... | Proof of (1). The condition is necessary because, if \( \widetilde{\varphi } \) is an extension of \( \varphi \) to \( G \), then, for \( h \in H \) and \( g \in G \) with \( {g}^{-1}{hg} \in H \), we have\n\n\[ \varphi \left( {{g}^{-1}{hg}}\right) = \widetilde{\varphi }{\left( g\right) }^{-1}\varphi \left( h\right) \w... | Yes |
Theorem 7.14. Let \( H \) be an abelian p-Sylow subgroup of \( G \) and let \( N \) be its normalizer in \( G \) . Then the image of \( \operatorname{Ver} : {G}^{\mathrm{{ab}}} \rightarrow {H}^{\mathrm{{ab}}} = H \) is \( {H}^{N} = H \cap Z\left( N\right) \) . | Proof. The group \( \operatorname{Im}\left( \operatorname{Ver}\right) \) is contained in \( {H}^{N} = \left\{ {h \in H \mid {xh}{x}^{-1} = h}\right. \), for all \( \left. {x \in N}\right\} \) , cf. prop.7.4.\n\nLet us show the opposite inclusion. Let \( C = {H}^{N} \) . If \( c \in C \) and \( h \in H \) are \( G \) -c... | Yes |
Theorem 7.15. Let H be a p-Sylow subgroup of \( G \) . Suppose that \( H \) is nontrivial abelian and that \( G \) has no cyclic quotient of order \( p \) . Let \( N \) be the normalizer of \( H \) in \( G \) . Then :\n\n(1) \( {H}^{N} = 1 \), where \( {H}^{N} \) is the set of elements of \( H \) fixed under \( N \) -c... | Proof of (1). Theorem 7.14 shows that Ver : \( G \rightarrow {H}^{N} \) is surjective. If \( {H}^{N} \neq 1 \), this implies that \( G \) has a quotient that is cyclic of order \( p \), which contradicts the assumption made on \( G \) . | Yes |
Corollary 7.16. We have \( {N}_{G}\left( H\right) \neq H \) . | Proof. If \( {N}_{G}\left( H\right) = H \), we have \( {H}^{N} = H \), contrary to (1). | No |
Corollary 7.17. If \( p = 2 \), the subgroup \( H \) is not cyclic. | Proof. Indeed, we have \( r \geq 2 \) by th.7.15 [for a transfer-free proof, see exerc. 11 of chap.2]. | No |
Corollary 7.19. A simple group of odd order \( < {2000} \) is cyclic. | Proof. Let \( G \) be a nonsolvable group of odd order \( N < {2000} \) . Write \( N \) as \( \prod {p}_{1}^{{m}_{1}}\cdots {p}_{n}^{{m}_{n}} \) , where the \( {p}_{i} \) are distinct primes and we have \( 1 < {p}_{1}^{{m}_{1}} < {p}_{2}^{{m}_{2}} < \cdots < {p}_{n}^{{m}_{n}} \) . According to Burnside’s theorem (cf. c... | Yes |
Lemma 7.21. Let \( G \) be a nonabelian simple group, and let \( A \) be a proper subgroup of \( G \) . Then \( \left( {G : A}\right) \geq 5 \) ; moreover \( \left( {G : A}\right) = 5 \) is impossible if \( \left| G\right| > {60} \) . | Proof of the lemma. If \( n = \left( {G : A}\right) \) is \( > 1 \), the action of \( G \) on \( G/A \) gives a nontrivial homomorphism \( G \rightarrow {\mathcal{S}}_{n} \), which is injective because \( G \) is simple. This is impossible if \( n = 2,3 \) or 4 since \( {\mathcal{S}}_{n} \) is solvable. When \( n = 5 \... | Yes |
Theorem 7.22. A simple group of order 60 is isomorphic to \( {\mathcal{A}}_{5} \) . | Proof. Let \( G \) be such a group. It is nonabelian since 60 is not a prime number. Let \( H \) be a 2-Sylow subgroup of \( G \) and let \( N \) be its normalizer. We have seen in part a) of the proof of th.7.20 that the order of \( N \) is divisible by 12, hence is equal to 12, otherwise \( N \) would be equal to \( ... | Yes |
Proposition 8.1. (1) \( {\chi }_{V} \) is a class function \( {}^{1} \) . | Proof. Formulas (1) to (4) follow from standard properties of traces; formula (5) follows from (3) and from \( \operatorname{Hom}\left( {{V}_{1},{V}_{2}}\right) \simeq {V}_{1}^{ * } \otimes {V}_{2} \) ; formula (6) is obvious. As for (7) and (8), note that, if the eigenvalues of \( {\rho }_{V}\left( g\right) \) are \( ... | Yes |
Proposition 8.2. The character of the permutation representation \( {V}_{X} \) is the function \( {\chi }_{X} \) of prop.1.1, namely the function :\n\n\[ \n{\chi }_{X}\left( g\right) = \left| {X}^{g}\right| = \text{ number of elements of }X\text{ fixed by }g.\n\]\n\n[When \( k \) has characteristic \( p > 0 \), the for... | Proof. The coefficient of \( {v}_{x} \) in \( g{v}_{x} \) is 1 if \( {gx} = x \) and 0 if \( {gx} \neq x \) . Hence the trace of the matrix representing \( g \) is the number of such 1’s, i.e., \( {\chi }_{X}\left( g\right) \) . | Yes |
Proposition 8.4. The endomorphism \( \rho \left( g\right) \) of \( V \) is diagonalizable (i.e., there exists a basis of \( V \) made up of eigenvectors of \( \rho \left( g\right) \) ). | Proof. If \( m \) is the order of \( g \), the minimal polynomial of \( \rho \left( g\right) \) divides \( {t}^{m} - 1 \) ; hence its roots are distinct, which implies (cf. e.g. Bourbaki [4], VII.7, prop.12) that \( \rho \left( g\right) \) is diagonalizable. | Yes |
Proposition 8.5. For every \( g \in G \) we have :\n\n(1) \( {\chi }_{V}\left( {g}^{-1}\right) = \overline{{\chi }_{V}\left( g\right) } \) .\n\n(2) \( \left| {{\chi }_{V}\left( g\right) }\right| \leq \dim V \) .\n\n(3) \( \left| {{\chi }_{V}\left( g\right) }\right| = \dim V \) if only if \( g \) acts on \( V \) by scal... | Proof. Let \( n = \dim V \), and let \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) be the eigenvalues of \( \rho \left( g\right) \) . Since \( \left| {\lambda }_{i}\right| = 1 \) for every \( i \), we have \( \left| {{\lambda }_{1} + \cdots + {\lambda }_{n}}\right| \leq n \), which proves (2). The eigenvalues of \( \rho ... | No |
Lemma 8.6. Let \( {z}_{1},\ldots ,{z}_{n} \) be complex numbers of absolute value 1. If \( \left| {{z}_{1} + \cdots + {z}_{n}}\right| = n \) , then \( {z}_{1} = {z}_{2} = \cdots = {z}_{n} \) . | Proof of the lemma. Let \( z = {z}_{1} + \cdots + {z}_{n} \) . Write \( {z}_{j} = {e}^{i{\varphi }_{j}} \) with \( {\varphi }_{j} \in \mathbf{R}/{2\pi }\mathbf{Z} \) . We have\n\n\[ z\bar{z} = \mathop{\sum }\limits_{{j,{j}^{\prime }}}{e}^{i\left( {{\varphi }_{j} - {\varphi }_{{j}^{\prime }}}\right) } = n + 2\mathop{\su... | Yes |
Corollary 8.7. We have \( {\chi }_{{V}^{ * }} = \overline{{\chi }_{V}} \) . | Proof. This follows from part (1) of prop.8.5 and part (4) of prop.8.1. | No |
Proposition 8.8. Let \( h \) be a positive definite hermitian form on \( V \) . Let \( W \) be a subspace of \( V \) and let \( {W}^{\prime } \) be its orthogonal for \( h \) (i.e., the set of \( x \in V \) such that \( h\left( {x, y}\right) = 0 \) for every \( y \in W \) ). Then \( {W}^{\prime } \) is a subspace of \(... | Proof. Let \( \left( {{e}_{1},\ldots ,{e}_{m}}\right) \) be a basis of \( W \) ; the set \( {W}^{\prime } \) is defined by the vanishing of the \( m \) linear forms \( x \mapsto h\left( {x,{e}_{i}}\right) \) ; hence it is a subvectorspace of dimension \( {m}^{\prime } \geq n - m \) , where \( n = \dim V \) . If \( x \i... | Yes |
Proposition 8.9. Let \( V \) be a linear representation of \( G \) . There exists a positive definite hermitian form on \( V \) that is invariant by \( G \) . | Proof. Choose a positive definite hermitian form \( {h}_{0} \) on \( V \), for instance the form \( {h}_{n} \) above, relative to a \( \mathbf{C} \) -isomorphism \( V \simeq {\mathbf{C}}^{n} \) . Define\n\n\[ h\left( {x, y}\right) = \mathop{\sum }\limits_{{g \in G}}{h}_{0}\left( {{gx},{gy}}\right) . \]\n\nIt is hermiti... | Yes |
Proposition 8.10. Let \( W \) be a \( G \) -invariant subspace of \( V \) .\n\n(1) There exists a G-invariant subspace of \( V \) complementary to \( W \) . | Proof. Choose a \( G \) -invariant positive definite hermitian form \( h \) on \( V \) . Let \( {W}^{\prime } \) be the orthogonal of \( W \) for \( h \) . By prop.8.8, we have \( V = W \oplus {W}^{\prime } \), and it is clear that \( {W}^{\prime } \) is \( G \) -invariant. This proves (1); as for (2), it follows from ... | Yes |
Theorem 8.11. Every representation is a direct sum of irreducible representations. | Proof. Use induction on \( \dim V \) . There is nothing to prove if \( \dim V \leq 1 \) . If \( \dim V > 1 \) , either \( V \) is irreducible (and we are done), or there exists a proper \( G \) -invariant subspace \( {V}^{\prime } \neq 0 \) of \( V \) . Then, by prop.8.10, there exists a direct sum decomposition of \( ... | Yes |
Theorem 8.12 (Schur’s lemma \( {}^{2},\left\lbrack {34}\right\rbrack ,{\mathrm{n}}^{ \circ }7 \) ). Let \( {\rho }_{1} : G \rightarrow \mathbf{{GL}}\left( {V}_{1}\right) \) and \( {\rho }_{2} : G \rightarrow \mathbf{{GL}}\left( {V}_{2}\right) \) be two irreducible representations of \( G \) . Let \( f : {V}_{1} \righta... | Proof. (1) If \( x \in \operatorname{Ker}f \), we have \( f\left( {{\rho }_{1}\left( g\right) }\right) \left( x\right) = {\rho }_{2}\left( g\right) \left( {f\left( x\right) }\right) = 0 \) for all \( g \in G \) . Thus, Ker \( f \) is \( G \) -invariant. Since \( {V}_{1} \) is irreducible, either Ker \( f = 0 \) or Ker ... | Yes |
Corollary 8.14. If \( G \) is abelian, its irreducible representations are 1-dimensional (i.e., they are given by homomorphisms \( G \rightarrow {\mathrm{{GL}}}_{1}\left( \mathbf{C}\right) = {\mathbf{C}}^{ \times } \) ). | Proof. Let \( V \) be an irreducible representation of \( G \) . By part (2) of Schur’s lemma, the elements of \( G \) act by homotheties; this implies that \( \dim V = 1 \) . | Yes |
Theorem 8.15 (Basic formula). Let \( V \) be a linear representation of \( G \), and let \( {V}^{G} \) be the subspace of \( V \) fixed by \( G \) . Then :\n\n\[ \dim {V}^{G} = \left\langle {{\chi }_{V},1}\right\rangle = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{g \in G}}{\chi }_{V}\left( g\right) . \] | Proof. Let \( \pi \) be the endomorphism \( x \mapsto \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{g \in G}}{gx} \) of \( V \) . We have \( {\pi x} \in {V}^{G} \) for every \( x \), and \( {\pi x} = x \) if \( x \in {V}^{G} \) ; this implies that \( {\pi }^{2} = \pi \), i.e., that \( \pi \) is an idempotent; its i... | Yes |
Proposition 8.16. Let \( \chi \) and \( {\chi }^{\prime } \) be the characters of two representations \( V \) and \( {V}^{\prime } \) of \( G \) . Then :\n\n\[ \left\langle {\chi ,{\chi }^{\prime }}\right\rangle = \dim {\operatorname{Hom}}_{G}\left( {{V}^{\prime }, V}\right) \] | Proof. View \( W = \operatorname{Hom}\left( {{V}^{\prime }, V}\right) \) as a representation of \( G \) . Its character is equal to \( \chi \overline{{\chi }^{\prime }} \) , cf. prop.8.1. An element \( f \) of \( W \) is fixed under \( G \) if and only if it is a \( G \) -homomorphism of \( {V}^{\prime } \) into \( V \... | Yes |
Corollary 8.17 (Orthogonality of irreducible characters). Suppose that \( V \) and \( {V}^{\prime } \) are irreducible. Then :\n\n\[ \left\langle {\chi ,{\chi }^{\prime }}\right\rangle = \left\{ \begin{array}{ll} 1 & \text{ if }V\text{ is isomorphic to }{V}^{\prime }, \\ 0 & \text{ otherwise. } \end{array}\right. \] | Proof. This follows from prop.8.16 and cor.8.13. | No |
Corollary 8.18. The irreducible characters of \( G \) are \( \mathbf{C} \) -linearly independent. | Proof. This follows from the orthogonality property of cor.8.17. | No |
Theorem 8.19. Let \( V \) be a representation with character \( {\chi }_{V} \) and let \( V = {\bigoplus }_{\alpha }{V}_{\alpha } \) be a splitting of \( V \) into a direct sum of irreducible representations \( {V}_{\alpha } \) . Let \( W \) be an irreducible representation of \( G \) . The number of \( {V}_{\alpha } \... | Proof. We have \( {\chi }_{V} = \mathop{\sum }\limits_{\alpha }{\chi }_{{V}_{\alpha }} \), hence \( \left\langle {{\chi }_{V},{\chi }_{W}}\right\rangle = \mathop{\sum }\limits_{\alpha }\left\langle {{\chi }_{{V}_{\alpha }},{\chi }_{W}}\right\rangle \) . By cor.8.17, \( \left\langle {{\chi }_{{V}_{\alpha }},{\chi }_{W}}... | Yes |
Corollary 8.20. Two representations with the same character are isomorphic. | Indeed, if we split the two representations in direct sums of irreducible ones, th.8.19 shows that each irreducible representation \( W \) occurs the same number of times. | No |
Proposition 8.21. Let \( \operatorname{Irr}\left( G\right) \) be the set of irreducible characters of \( G \) . Then :\n\n(1) \( {r}_{G} = \mathop{\sum }\limits_{{\chi \in \operatorname{Irr}\left( G\right) }}\chi \left( 1\right) \chi \) . | Proof. The values of \( {r}_{G}\left( g\right) \) given in cor. \( {8.3} \) show that \( \left\langle {\chi ,{r}_{G}}\right\rangle = \chi \left( 1\right) \) ; this proves formula (1). | Yes |
Theorem 8.22. The map \( r : \mathbf{C}\left\lbrack G\right\rbrack \rightarrow \mathop{\prod }\limits_{{\chi \in \operatorname{Irr}\left( G\right) }}\operatorname{End}\left( {W}_{\chi }\right) \) is an isomorphism. | Proof. By part (2) of prop.8.21, we have \( \left| G\right| = \sum \chi {\left( 1\right) }^{2} \) ; this shows that the two algebras have the same dimension. Hence we only have to prove that \( r \) is injective, which is clear: an element of \( \operatorname{Ker}\left( r\right) \) acts by 0 on every linear representat... | Yes |
Theorem 8.23. Let \( \left( {u}_{\chi }\right) \) be an element of \( \mathop{\prod }\limits_{{\chi \in \operatorname{Irr}}}\operatorname{End}\left( {W}_{\chi }\right) \) . If \( g \in G \), put\n\n\[ \n{\lambda }_{g} = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{\chi \in \operatorname{Irr}\left( G\right) }}\chi ... | Proof. Let us consider first the case where \( {u}_{\chi } = {\rho }_{\chi }\left( x\right) \), where \( x \) is a given element of \( G \) . In that case, we have\n\n\[ \n{\lambda }_{g} = \frac{1}{\left| G\right| }\mathop{\sum }\limits_{\chi }\chi \left( 1\right) \chi \left( {{g}^{-1}x}\right) = \frac{1}{\left| G\righ... | Yes |
Theorem 8.24. The isomorphism \( r : \mathbf{C}\left\lbrack G\right\rbrack \rightarrow \mathop{\prod }\limits_{\chi }\operatorname{End}\left( {W}_{\chi }\right) \) of th.8.22 induces an isomorphism \( \omega : \mathbf{C}{\left\lbrack G\right\rbrack }^{\text{cent }} \rightarrow {\mathbf{C}}^{\operatorname{Irr}\left( G\r... | Proof. This follows from th.8.22, and the fact that the center of \( \operatorname{End}\left( {W}_{\chi }\right) \) is \( \mathbf{C} \) . | Yes |
Corollary 8.25. \( \left| {\operatorname{Irr}\left( G\right) }\right| = h \), i.e., the number of irreducible characters of \( G \) is equal to the number of conjugacy classes of \( G \) . | Proof. This follows from \( \dim \mathbf{C}{\left\lbrack G\right\rbrack }^{\text{cent }} = h \) and \( \dim {\mathbf{C}}^{\operatorname{Irr}\left( G\right) } = \left| {\operatorname{Irr}\left( G\right) }\right| \) . | Yes |
Proposition 8.28. For every element \( \mathop{\sum }\limits_{{g \in G}}f\left( g\right) g \) of \( \mathbf{C}{\left\lbrack G\right\rbrack }^{\text{cent }} \), we have :\n\n\[ \n{\omega }_{\chi }\left( {\mathop{\sum }\limits_{{g \in G}}f\left( g\right) g}\right) = \frac{1}{\chi \left( 1\right) }\mathop{\sum }\limits_{{... | Proof. By definition, \( {\omega }_{\chi }\left( {\mathop{\sum }\limits_{g}f\left( g\right) g}\right) \) is the scalar factor of the homothety of \( {W}_{\chi } \) given by \( \mathop{\sum }\limits_{g}f\left( g\right) g \) ; hence it is equal to the trace of that homothety divided by \( \chi \left( 1\right) \) ; since ... | Yes |
Proposition 8.29. We have\n\n\[ \n{\varepsilon }_{\chi } = \frac{\chi \left( 1\right) }{\left| G\right| }\mathop{\sum }\limits_{{g \in G}}\chi \left( {g}^{-1}\right) g.\n\] | Proof. The image of \( {\varepsilon }_{\chi } \) in \( \operatorname{End}\left( {W}_{\chi }\right) \) is 1, and its image in \( \operatorname{End}\left( {W}_{\psi }\right) \) is 0 if \( \psi \neq \chi \) . Equation (8.6), applied to \( {u}_{\chi } = 1 \) and \( {u}_{\psi } = 0 \) if \( \psi \neq \chi \), gives (8.8). | No |
Proposition 8.30. The following properties are equivalent :\n\n(1) There exist an integer \( n \geq 1 \), and elements \( {a}_{1},\ldots ,{a}_{n} \) of \( \mathbf{Z} \) such that :\n\n\[ {x}^{n} + {a}_{1}{x}^{n - 1} + \cdots + {a}_{n} = 0. \]\n\n(8.11)\n\n(2) The subring \( \mathbf{Z}\left\lbrack x\right\rbrack \) of \... | Proof. Equation (8.11) implies that \( \mathbf{Z}\left\lbrack x\right\rbrack \) is additively generated by \( 1, x,\ldots ,{x}^{n - 1} \) ; hence (1) implies (2). Conversely, suppose that (2) holds; for every \( i \geq 0 \), let \( {L}_{i} \) be the \( \mathbf{Z} \) - module generated by \( 1, x,\ldots ,{x}^{i} \) ; we... | Yes |
Proposition 8.31. The set of \( \mathbf{Z} \) -integral elements of \( R \) is a subring of \( R \) . | Proof. Let \( x, y \) be \( \mathbf{Z} \) -integral elements of \( R \) . Then the ring \( P = \mathbf{Z}\left\lbrack x\right\rbrack { \otimes }_{\mathbf{Z}}\mathbf{Z}\left\lbrack y\right\rbrack \) is finitely generated over \( \mathbf{Z} \) . The subring \( \mathbf{Z}\left\lbrack {x, y}\right\rbrack \) of \( R \) gene... | Yes |
Proposition 8.32. An element of \( \mathbf{Q} \) is \( \mathbf{Z} \) -integral if and only if it belongs to \( \mathbf{Z} \) . | Proof. If \( x \in \mathbf{Q} \) is \( \mathbf{Z} \) -integral, it satisfies an equation of type (8.11):\n\n\[ {x}^{n} + {a}_{1}{x}^{n - 1} + \cdots + {a}_{n} = 0,\text{ with }{a}_{i} \in \mathbf{Z}. \]\n\nIf we write \( x \) as \( p/q \), with \( \left( {p, q}\right) \) relatively prime, this equation gives\n\n\[ {p}^... | Yes |
Lemma 8.33. Let \( {x}_{1},\ldots ,{x}_{n} \in \mathbf{C} \) be roots of unity. Suppose that \( x = \left( {{x}_{1} + \cdots + {x}_{n}}\right) /n \) is an algebraic integer. Then, either \( x = 0 \), or \( {x}_{1} = \cdots = {x}_{n} \) . | Proof. Choose a finite Galois extension \( K/\mathbf{Q} \), contained in \( \mathbf{C} \), and containing all the \( {x}_{i} \) ; let \( \Gamma = \operatorname{Gal}\left( {K/\mathbf{Q}}\right) \) . Since \( x \) is \( \mathbf{Z} \) -integral, the same is true of all the \( \gamma \left( x\right) ,\gamma \in \Gamma \), ... | Yes |
Proposition 8.34. The values of a character are algebraic integers. | Proof. Indeed, they are sums of roots of unity. | No |
Theorem 8.35. Let \( f \) be a class function on \( G \) such that \( f\left( g\right) \) is an algebraic integer for every \( g \in G \). (1) The element \( x = \mathop{\sum }\limits_{g}f\left( g\right) g \) of \( \mathbf{C}{\left\lbrack G\right\rbrack }^{\text{cent }} \) is \( \mathbf{Z} \) -integral. | Proof of (1). Let \( R = \mathbf{C}{\left\lbrack G\right\rbrack }^{\text{cent }} \), and let \( {R}_{\mathbf{Z}} \) be the subring of \( R \) made up of the \( \mathop{\sum }\limits_{g}\varphi \left( g\right) g \) where \( \varphi \) is a class function with values in \( \mathbf{Z} \) . If \( \sigma \) is a conjugacy c... | Yes |
Corollary 8.36. If \( \chi \in \operatorname{Irr}\left( G\right) \), then \( \chi \left( 1\right) \) divides \( \left| G\right| \) . | Proof. Part (2) of th.8.35, applied to \( f = \bar{\chi } \), shows that \( \left| G\right| /\chi \left( 1\right) \) is an algebraic integer; by prop.8.32, this means that \( \chi \left( 1\right) \) divides \( \left| G\right| \) . | Yes |
Proposition 8.37. Let \( \chi \) be an irreducible character, and let \( {\rho }_{\chi } \) be a linear representation corresponding to \( \chi \) . Let \( g \in G \) and let \( c\left( g\right) \) be the number of elements of the conjugacy class of \( g \) . Then :\n\n(1) \( c\left( g\right) \chi \left( g\right) /\chi... | Proof of (1). Let \( f \) be the class function on \( G \) which is equal to 1 on the class of \( g \), and vanishes elsewhere. We have \( \mathop{\sum }\limits_{{z \in G}}f\left( z\right) \chi \left( z\right) = c\left( g\right) \chi \left( g\right) \) . Part (2) of th.8.35 shows that \( c\left( g\right) \chi \left( g\... | Yes |
Theorem 8.38 (Irreducibility of the cyclotomic polynomial). The map \( a : \sigma \mapsto a\left( \sigma \right) \) is an isomorphism of \( {\sum }_{N} \) onto \( {\left( \mathbf{Z}/N\mathbf{Z}\right) }^{ \times } \) . | Proof. The fact that \( a \) is injective is clear. Its surjectivity was proved in 1801 by Gauss when \( N \) is a prime number, and in 1854 by Kronecker in the general case. Proofs can be found in Lang [31], chap.6, th.3.1 and in Bourbaki [4], chap.V, §11, th.2. | No |
Corollary 8.40. For every integer \( m \) prime to \( N \), there exists an automorphism of \( \mathbf{C} \) which coincides with \( {\sigma }_{m} \) on \( {K}_{N} \) . | Proof. Since \( \mathbf{C} \) is algebraically closed, every automorphism of \( {K}_{N} \) can be extended to an automorphism of \( \mathbf{C} \), cf. Bourbaki [4], chap.V, \( §{14} \), cor. 1 to prop.8. | Yes |
Theorem 8.41. (1) The function \( {\sigma }_{m}\left( \chi \right) \) is a character of \( G \). (2) \( \chi \in \operatorname{Irr}\left( G\right) \Rightarrow {\sigma }_{m}\left( \chi \right) \in \operatorname{Irr}\left( G\right) \). (3) \( {\sigma }_{m}\left( \chi \right) \left( g\right) = \chi \left( {g}^{m}\right) \... | Proof. Let \( \rho : G \rightarrow {\mathrm{{GL}}}_{n}\left( \mathbf{C}\right) \) be a linear representation of \( G \) with character \( \chi \). Let \( \sigma \in \operatorname{Aut}\left( \mathbf{C}\right) \) be an automorphism of \( \mathbf{C} \) extending \( {\sigma }_{m} \), cf. cor.8.40, and let \( {\sigma }^{\pr... | Yes |
Proposition 8.42. Let \( g \in G \) and let \( {\sum }_{N, g} \) be the subgroup of \( {\sum }_{N} \) made up of the \( m \) such that \( {g}^{m} \) is conjugate to \( g \) . Let \( {K}_{g} \) be the subfield of \( {K}_{N} \) fixed under the action of \( {\sum }_{N, g} \) . Then :\n\n(1) We have \( \chi \left( g\right)... | Proof. Let \( {K}_{g}^{\prime } \) be the subfield of \( {K}_{N} \) generated by all the \( \chi \left( g\right) \) . An element \( {\sigma }_{m} \) of \( {\sum }_{N} \) fixes \( {K}_{g}^{\prime } \) if and only if if fixes all the \( \chi \left( g\right) \) ; by part (3) of th.8.41, this means that \( \chi \left( g\ri... | Yes |
Proposition 8.43. Let \( \Phi \) be an abelian group. For every object \( V \) of \( {\mathcal{C}}_{G} \), let \( \phi \left( V\right) \) be an element of \( \Phi \) ; assume that \( \phi \left( V\right) = \phi \left( {V}^{\prime }\right) + \phi \left( {V}^{\prime \prime }\right) \) for every exact sequence\n\n\[ 0 \ri... | Proof. Indeed, \( {\iota }_{\phi } \) is characterized by its value for \( \chi \in \operatorname{Irr}\left( G\right) \), which is \( \phi \left( {V}_{\chi }\right) \) . | No |
Proposition 8.44. With the notation of \( §{8.7} \), let \( {R}^{\prime }\left( G\right) \) be the ring of class functions on \( G \) with values in \( {A}_{N} \) such that \( f\left( {g}^{m}\right) = {\sigma }_{m}\left( {f\left( g\right) }\right) \) for every \( g \in G \) and every integer \( m \) prime to \( \left| ... | Proof of (1). This follows from part (3) of th.8.41.\n\nProof of (2). Let us write \( \left| G\right| \) . \( f \) as a linear combination \( \sum {c}_{\chi }\chi \) of irreducible characters. We need to prove that \( {c}_{\chi } \) belongs to \( \mathbf{Z} \) for every \( \chi \) . We have \( {c}_{\chi } = \mathop{\su... | Yes |
Corollary 8.45. \( R\left( G\right) \) has finite index in \( {R}^{\prime }\left( G\right) \) . | Proof. This follows from (1) and (2). | No |
Theorem 8.46. If \( f \in R\left( G\right) \), then \( {\Psi }^{m}f \in R\left( G\right) \) for every \( m \in \mathbf{Z} \) . | Proof. The formulas above show that it is enough to prove this when \( m \) is \( > 0 \) . Since \( {\Psi }^{m} \) is linear, we may also assume that \( f \) is the character of a linear representation \( \rho \) of \( G \) ; let \( n = f\left( 1\right) \) be the degree of \( \rho \) . Let \( P\left( {{X}_{1},\ldots ,{... | No |
Theorem 8.47. We have \( {f}_{P} \in R\left( G\right) \) if the coefficients of \( P \) belong to \( \mathbf{Z} \) . | Proof of theorem 8.47. Consider first the case where \( P \) is an elementary symmetric polynomial \( {S}_{k} = \mathop{\sum }\limits_{{{i}_{1} < \cdots < {i}_{k}}}{X}_{{i}_{1}}\cdots {X}_{{i}_{k}} \) . In that case, \( {f}_{P}\left( g\right) \) is the trace of \( { \land }^{k}\rho \left( g\right) \), i.e., \( {f}_{P} ... | Yes |
Lemma 8.49. Let \( E \) be a vector space of dimension \( n \) over a field \( F \) . If \( u \in \operatorname{End}\left( E\right) \) has rank \( r \), then \( \dim \operatorname{End}\left( E\right) u = {nr} \) . | Proof of lemma 8.49. Let \( N \subset E \) be the kernel of \( u \) . Every element \( v \) of the ideal \( \operatorname{End}\left( E\right) u \) vanishes on \( N \) . Conservely, if \( v \) has that property, it can be factored as \( v = w \circ {u}^{\prime } \), where \( {u}^{\prime } : E \rightarrow \operatorname{I... | No |
Corollary 8.50. Suppose that there exist \( g \in G \) and \( \lambda \in k \) such that \( \lambda \) is an eigenvalue of \( {g}_{V} \) of multiplicity 1 . Then \( V \) is realizable over \( k \) . | Proof. This follows from th.8.48 applied to \( a = g - \lambda \) . [Note that the proof gives an explicit construction of a \( k \) -realization of \( V \), namely the \( k\left\lbrack G\right\rbrack \) - submodule of \( V \) generated by a nonzero eigenvector of \( {g}_{V} \) with eigenvalue \( \lambda \) .] | Yes |
Theorem 8.51. A linear representation \( V \) of \( G \) is realizable over \( \mathbf{R} \) if and only if there exists a nondegenerate symmetric bilinear form on \( V \) which is \( G \) -invariant. | Proof. If \( V \) is of the form \( {V}_{0}{ \otimes }_{\mathbf{R}}\mathbf{C} \), where \( {V}_{0} \) is an \( \mathbf{R} \) -representation of \( G \), let \( {B}_{0}\left( {x, y}\right) \) be a symmetric \( \mathbf{R} \) -bilinear form on \( {V}_{0} \), which is positive definite, i.e., such that \( {B}_{0}\left( {x,... | Yes |
Proposition 8.52. Let \( E \) be a finite dimensional vector space over \( \mathbf{C} \). Let \( h\left( {x, y}\right) \) be a positive definite hermitian form on \( E \) and let \( B\left( {x, y}\right) \) be a nondegenerate symmetric bilinear form on \( E \). Then there exists an \( \mathbf{R} \) -vector subspace \( ... | Proof of prop.8.52.\n\nLet us write \( h\left( {x, y}\right) \) as \( \langle x, y\rangle \). For every \( y \in E \), the map \( x \mapsto B\left( {x, y}\right) \) is linear; hence there exists \( \varphi \left( x\right) \in E \) such that \( B\left( {x, y}\right) = \langle \varphi \left( x\right), y\rangle \). The ma... | Yes |
Theorem 8.53. Let \( V \) be an irreducible representation of \( G \), and let \( \chi \) be its character. There are three different possibilities :\n\n(1) \( \left\langle {{\Psi }^{2}\chi ,1}\right\rangle = 0 \) ; the character \( \chi \) is not real valued; the only \( G \) -invariant bilinear form on \( V \) is 0 .... | Proof. If \( \chi \neq \bar{\chi }, V \) is not isomorphic to its dual \( {V}^{ * } \), hence \( {\operatorname{Hom}}_{G}\left( {V,{V}^{ * }}\right) = 0 \), which means that the only \( G \) -invariant bilinear form over \( V \) is 0 . We have \( \langle \chi ,\bar{\chi }\rangle = 0 \), i.e., \( \left\langle {{\chi }^{... | Yes |
Theorem 8.54. The set \( N \) is a subgroup of \( G \) . | Strategy of the proof. The main step consists in showing that every linear representation of \( H \) can be extended to \( G \) in such a way that it is trivial on \( N \) ; this will be done by extending the character of the representation from \( H \) to \( G \) and proving that the function so obtained is a characte... | No |
Lemma 8.55. Let \( f \) be a class function on \( H \) . There exists a unique class function \( \widetilde{f} \) on \( G \) which extends \( f \) and is constant on \( N \) . | Proof. The uniqueness of \( \widetilde{f} \) follows from the fact that every conjugacy class of \( G \) either meets \( H \), or is contained in \( N \) . Let us prove the existence. If \( x \in N \), we put \( \widetilde{f}\left( x\right) = f\left( 1\right) \) . If \( x \notin N \), we write \( x \) in the form \( {g... | Yes |
Lemma 8.56. Let \( f \) and \( \widetilde{f} \) be as in lemma 8.55, and let \( \theta \) be a class function on \( G \) ; let \( {\theta }_{H} \) be the restriction of \( \theta \) to \( H \) . Then :\n\n\[ \langle \widetilde{f},\theta {\rangle }_{G} = {\left\langle f,{\theta }_{H}\right\rangle }_{H} + f\left( 1\right... | Proof. Equation (8.22) holds if \( f = 1 \), since then \( \widetilde{f} = 1 \) . Hence, we only need to prove it when \( f\left( 1\right) = 0 \), in which case the formula reduces to :\n\n\[ \langle \widetilde{f},\theta {\rangle }_{G} = {\left\langle f,{\theta }_{H}\right\rangle }_{H} \]\n\nLet \( \mathcal{R} \) be a ... | Yes |
Lemma 8.57. The map \( f \mapsto \widetilde{f} \) is an isometry, i.e., \( {\left\langle {\widetilde{f}}_{1},{\widetilde{f}}_{2}\right\rangle }_{G} = {\left\langle {f}_{1},{f}_{2}\right\rangle }_{H} \) for every class functions \( {f}_{1} \) and \( {f}_{2} \) on \( G \) . | Proof. Equation (8.22), applied to \( \theta = 1 \), shows that \( \langle \widetilde{f},1{\rangle }_{G} = \langle f,1{\rangle }_{H} \) . Define \( {f}_{2}^{ * } \) by \( {f}_{2}^{ * }\left( g\right) = {f}_{2}\left( {g}^{-1}\right) \) . We have : \[ {\left\langle {\widetilde{f}}_{1},{\widetilde{f}}_{2}\right\rangle }_{... | Yes |
Lemma 8.58. If \( f \) is a character of \( H \) and \( \theta \) is a character of \( G \), then \( \langle \widetilde{f},\theta {\rangle }_{G} \) is an integer. | Proof. Since \( \theta \) is a character of \( G \), its restriction \( {\theta }_{H} \) to \( H \) is a character of \( H \), hence every term on the right side of equation (8.22) is an integer. | No |
Lemma 8.59. If \( \chi \) is an irreducible character of \( H \), then \( \widetilde{\chi } \) is an irreducible character of \( G \) . | Proof. Let \( {\theta }_{1},\ldots ,{\theta }_{m} \) be the irreducible characters of \( G \) . Since \( \widetilde{\chi } \) is a class function, we have \( \widetilde{\chi } = \sum {c}_{i}{\theta }_{i} \), with \( {c}_{i} \in \mathbf{C} \) . Lemma 8.58 shows that the \( {c}_{i} \) are integers. We have \( \sum {c}_{i... | Yes |
Lemma 8.60. If \( \chi \) is a character of \( H \), then \( \widetilde{\chi } \) is a character of \( G \) . | Proof. This follows from lemma 8.59 by writing \( \chi \) as a sum of irreducible characters. | No |
Proposition 8.61. Let \( g \) be an element of \( G - \{ 1\} \) and let \( c\left( g\right) \) be the number of elements of the conjugacy class of \( g \) . Suppose that \( c\left( g\right) \) is a power of a prime \( p \) . Then there exists a normal subgroup \( N \) of \( G \), different from \( G \), such that the i... | Proof. By part (3) of th.8.21 we have \( \mathop{\sum }\limits_{{\chi \in \operatorname{Irr}\left( G\right) }}\chi \left( 1\right) \chi \left( g\right) = 0 \) . Therefore,\n\n\[ \mathop{\sum }\limits_{{\chi \neq 1}}\frac{\chi \left( 1\right) \chi \left( g\right) }{p} = - \frac{1}{p} \]\n\nSince \( - \frac{1}{p} \) is n... | Yes |
Theorem 8.62 (Burnside). Every group \( G \) of order \( {p}^{a}{q}^{b} \) (where \( p \) and \( q \) are primes) is solvable. | Proof. Use induction on \( \left| G\right| \) . We may assume that \( G \) is a simple group, and also that \( b \) is nonzero (otherwise \( G \) is a \( p \) -group). If \( g \in G \), denote by \( c\left( g\right) \) the number of elements of the conjugacy class of \( g \) . There exists \( g \in G - \{ 1\} \) such t... | Yes |
Theorem 9.1. (1) The order of a finite subgroup of \( {\mathrm{{GL}}}_{n}\left( \mathbf{Q}\right) \) divides \( M\left( n\right) \) . (2) \( M\left( n\right) \) is the smallest integer having property (1). | Minkowski's proof of part (1) consisted in proving :\n\n(a) If \( G \) is a finite subgroup of \( {\mathrm{{GL}}}_{n}\left( \mathbf{Q}\right) \), then \( \left| G\right| \) divides \( \left| {{\mathrm{{GL}}}_{n}\left( {\mathbf{F}}_{p}\right) }\right| \) for every prime \( p > 2 \) .\n\n(b) If \( \ell > 2 \), there are ... | No |
Lemma 9.2. Assertion (1) is equivalent to :\n\n\( \left( {1}^{\prime }\right) \) If \( \ell \) is a prime number, and \( A \) is a finite \( \ell \) -subgroup of \( {\mathrm{{GL}}}_{n}\left( \mathbf{Q}\right) \), then \( \left| A\right| \leq {\ell }^{M\left( {n,\ell }\right) } \) . | Proof. \( \left( 1\right) \Rightarrow \left( {1}^{\prime }\right) \) is clear. The converse follows from Sylow’s theorem. | No |
Lemma 9.3. If \( p \) is large enough, the homomorphism\n\n\[ A \rightarrow {\mathrm{{GL}}}_{n}\left( {\mathbf{Z}\left\lbrack {1/q}\right\rbrack }\right) \rightarrow {\mathrm{{GL}}}_{n}\left( {\mathbf{F}}_{p}\right) \]\n\nis injective. | Proof. For every \( a \in A - \{ 1\} \), choose a coefficient \( {x}_{a} \) of the matrix \( q\left( {a - 1}\right) \) which is a nonzero integer. If \( p \) does not divide \( q\mathop{\prod }\limits_{{a \neq 1}}{x}_{a} \), the kernel of \( A \rightarrow {\mathrm{{GL}}}_{n}\left( {\mathbf{F}}_{p}\right) \) is trivial. | Yes |
Lemma 9.4. If \( p \) is large enough, the order of \( A \) divides \( {r}_{n}\left( p\right) = \mathop{\prod }\limits_{{i = 1}}^{n}\left( {{p}^{i} - 1}\right) \) . | Proof. By lemma 9.3, \( A \) is isomorphic to a subgroup of \( {\mathrm{{GL}}}_{n}\left( {\mathbf{F}}_{p}\right) \) . Hence its order divides the order of \( {\mathrm{{GL}}}_{n}\left( {\mathbf{F}}_{p}\right) \), which is equal to \( {p}^{n\left( {n - 1}\right) /2}{r}_{n}\left( p\right) \) . Since \( A \) is an \( \ell ... | Yes |
Lemma 9.5. Let \( m \) be an integer \( > 0 \) . Then :\n\n\[ \n{v}_{\ell }\left( {m!}\right) = \left\lbrack \frac{m}{\ell }\right\rbrack + \left\lbrack \frac{m}{{\ell }^{2}}\right\rbrack + \left\lbrack \frac{m}{{\ell }^{3}}\right\rbrack + \cdots \n\] | Proof. For every \( j \), let \( {a}_{j} \) be the number of \( z \in \left\lbrack {1, m}\right\rbrack \) such that \( {v}_{\ell }\left( z\right) = j \) . We have \( {v}_{\ell }\left( {m!}\right) = \mathop{\sum }\limits_{{z = 1}}^{m}{v}_{\ell }\left( z\right) = \mathop{\sum }\limits_{j}j{a}_{j} \), and \( {a}_{j} = {b}... | Yes |
Lemma 9.6. Let \( \\left( {x, a, b}\\right) \) be integers such that \( a, b \\geq 1 \) . Then \( \\left\\lbrack {\\left\\lbrack {x/a}\\right\\rbrack /b}\\right\\rbrack = \\left\\lbrack {x/{ab}}\\right\\rbrack \) . | Proof. We have \( x = {ay} + \\epsilon \) with \( y = \\left\\lbrack {x/a}\\right\\rbrack \) and \( 0 \\leq \\epsilon \\leq a - 1 \) . If \( z = \\left\\lbrack {y/b}\\right\\rbrack \), we have \( y = {bz} + \\eta \) with \( 0 \\leq \\eta \\leq b - 1 \) . Hence \( x = {abz} + {a\\eta } + \\epsilon \\leq {abz} + a\\left(... | Yes |
Lemma 9.7. Let \( x \in \mathbf{Z} \) be such that the class of \( x{\;\operatorname{mod}\;{\ell }^{2}} \) generates \( {\left( \mathbf{Z}/{\ell }^{2}\mathbf{Z}\right) }^{ \times } \) . Let \( k \) be an integer \( \geq 1 \) . Then :\n\n\[ \n{v}_{\ell }\left( {{x}^{k} - 1}\right) = \left\{ \begin{array}{ll} 0 & \text{ ... | Proof. If \( \ell - 1 \) does not divide \( k \), then \( {x}^{k} ≢ 1\left( {\;\operatorname{mod}\;\ell }\right) \) and \( {v}_{\ell }\left( {{x}^{k} - 1}\right) = 0 \) .\n\nIf \( k = \left( {\ell - 1}\right) m \) with \( m \geq 1 \), we have \( {x}^{k} = {y}^{m} \), where \( y = {x}^{\ell - 1} \) . The hypothesis made... | Yes |
Lemma 9.8. Let \( x \in \mathbf{Z} \) be as in lemma 9.7. Then\n\n\[ \n{v}_{\ell }\left( {\mathop{\prod }\limits_{{i = 1}}^{n}\left( {{x}^{i} - 1}\right) }\right) = M\left( {n,\ell }\right) .\n\] | Proof. Let \( N = \mathop{\prod }\limits_{{i = 1}}^{n}\left( {{x}^{i} - 1}\right) \) . We have \( {v}_{\ell }\left( N\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{v}_{\ell }\left( {{x}^{i} - 1}\right) \) . By lemma 9.7, \( {v}_{\ell }\left( {{x}^{i} - 1}\right) = 0 \) if \( i \) is not divisible by \( \ell - 1 \) ; the... | Yes |
Theorem 9.9. For every \( n \geq 1 \), there exists a real number \( f\left( n\right) \) such that every finite subgroup of \( {\mathrm{{GL}}}_{n}\left( \mathbf{C}\right) \) contains an abelian normal subgroup of index \( \leq f\left( n\right) \) . | We shall prove th.9.9 in \( §{9.2.6} \) by using a method of Frobenius \( \left( {\left\lbrack {19}\right\rbrack ,{\mathrm{n}}^{ \circ }{87}}\right) \) which gives an explicit value for \( f\left( n\right) \), namely : \[ f\left( n\right) = {\left( \sqrt{8n} + 1\right) }^{2{n}^{2}} - {\left( \sqrt{8n} - 1\right) }^{2{n... | Yes |
Proposition 9.10. Let \( a \in M \) be normal. Then :\n\n(1) If \( x \in V \) is such that \( {ax} = {\lambda x},\lambda \in \mathbf{C} \), then \( {a}^{ * }x = \bar{\lambda }x \) ; if \( y \in V \) is such that \( \langle x, y\rangle = 0 \), then \( \langle x,{ay}\rangle = 0 \) and \( \left\langle {x,{a}^{ * }y}\right... | Proof of (1). After replacing \( a \) by \( a - \lambda \) and \( {a}^{ * } \) by \( {a}^{ * } - \bar{\lambda } \), we may assume that \( \lambda = 0 \) . We then have \( \left\langle {{a}^{ * }x,{a}^{ * }x}\right\rangle = \left\langle {x, a{a}^{ * }x}\right\rangle = \left\langle {x,{a}^{ * }{ax}}\right\rangle = 0 \), ... | Yes |
Corollary 9.11. Let \( a \in M \) be positive hermitian and let \( m \) be an integer \( \geq 1 \) . There exists a unique positive hermitian \( b \) such that \( a = {b}^{m} \) . | Proof. Let \( L \) be the set of eigenvalues of \( a \) ; since \( a \) is positive hermitian, the elements of \( L \) are real \( \geq 0 \), and \( E \) is the orthogonal direct sum \( E = { \oplus }_{\lambda \in L}{E}_{\lambda } \) of the corresponding eigenspaces \( {E}_{\lambda } \), cf. prop.9.10. The automorphism... | Yes |
Proposition 9.12. (1) \( N\left( 1\right) = n \) . | Proof of (1). It follows from \( \operatorname{Tr}\left( 1\right) = n \) . | No |
Proposition 9.13. Let \( u, v \in M \) be unitary, and let \( c = {uv}{u}^{-1}{v}^{-1} \). Suppose that \( u \) and \( c \) commute and that \( N\left( {1 - v}\right) < 2 \). Then \( u \) and \( v \) commute. [The condition \( N\left( {1 - v}\right) < 2 \) can be weakened to \( N\left( {1 - v}\right) < 4 \), which is b... | Proof. Let \( {u}^{\prime } = {u}^{-1}c = {vu}{v}^{-1} \). Since \( u \) and \( c \) commute, the same is true for \( u \) and \( {u}^{\prime } \). By applying prop.9.10 to \( u \), and then to \( {u}^{\prime } \), we may find an orthonormal basis \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) made up of eigenvectors of... | Yes |
Proposition 9.14. Let \( u \) and \( v \) be elements of a finite subgroup of the unitary group \( {\mathbf{U}}_{V} \) . If \( N\left( {1 - u}\right) < 1/2 \) and \( N\left( {1 - v}\right) < 2 \), then \( u \) and \( v \) commute. | Proof. Define \( {v}_{0},{v}_{1},\ldots \) by induction, with \( {v}_{0} = v \) and \( {v}_{m} = u{v}_{m - 1}{u}^{-1}{v}_{m - 1}^{-1} \) for \( m \geq 1 \) . By part (6) of prop.9.12, we have\n\n\[ N\left( {1 - {v}_{m}}\right) \leq {2}^{m}N{\left( 1 - u\right) }^{m}N\left( {1 - v}\right) < 2. \]\n\n(9.9)\n\nSince \( {2... | Yes |
Corollary 9.15. Let \( U \) be the set of all \( u \in {\mathbf{U}}_{V} \) such that \( N\left( {1 - u}\right) < 1/2 \) . Let \( G \) be a finite subgroup of \( {\mathbf{U}}_{V} \) and let \( A \) be the subgroup of \( G \) generated by \( G \cap U \) . Then \( A \) is a normal abelian subgroup of \( G \) . | Proof. That \( A \) is normal is clear, since \( U \) is stable under the inner automorphisms of \( {\mathbf{U}}_{V} \) , cf. part (2) of prop.9.12. If \( u, v \) belong to \( G \cap U \), prop.9.14 shows that they commute. Hence \( A \) is abelian. | Yes |
Proposition 9.16. Let \( \delta \) be a real number, with \( 0 < \delta < 1 \) . Let \( X \subset {\mathbf{S}}_{d - 1} \) be such that \( x, y \in X, x \neq y \), implies \( \left| {x - y}\right| \geq {2\delta } \) . Then :\n\n\[ \left| X\right| \leq {\left( \frac{1}{\delta } + 1\right) }^{d} - {\left( \frac{1}{\delta ... | Proof. For every \( x \in X \), let \( {B}_{x} \) be the set of \( y \) such that \( \left| {x - y}\right| < \delta \) . The \( {B}_{x} \) are open balls, and they are disjoint. They are contained in the open ball \( {B}_{ + } \) made up of the \( y \) such that \( \left| y\right| < 1 + \delta \), and they do not inter... | Yes |
Proposition 10.1. Suppose that \( q = {2}^{m} \) . Then :\n\n(1) The group \( {\mathcal{S}}_{4} \) cannot be embedded in \( {\mathrm{{PGL}}}_{2}\left( {\mathbf{F}}_{q}\right) \) . | Proof of (1). The 2-Sylow subgroups of \( {\mathrm{{PGL}}}_{2}\left( {\mathbf{F}}_{q}\right) \) are abelian, and those of \( {\mathcal{S}}_{4} \) are not. | Yes |
Proposition 10.2. Assume that \( q \) is not a power of 2. Then :\n\n(1) The group \( {\mathcal{A}}_{4} \) can be embedded in \( {\operatorname{PSL}}_{2}\left( {\mathbf{F}}_{q}\right) \) . | - If \( q ≢ \pm 1\left( {\;\operatorname{mod}\;8}\right) \), the order of \( {\operatorname{PSL}}_{2}\left( {\mathbf{F}}_{q}\right) \), which is \( \frac{1}{2}q\left( {{q}^{2} - 1}\right) \), is not divisible by 8, hence that group does not contain \( {\mathcal{S}}_{4} \) whose order is 24 . | No |
Simplify: | Distribute -2 and then combine like terms. | No |
An object is launched from the ground at a speed of 64 feet per second. Write a function that models the height of the object and use it to calculate the objects height at 1 second and at 3.5 seconds. | Solution:\n\nWe know that the acceleration due to gravity is \( g = {32} \) feet per second squared and we are given the initial velocity \( {v0} = {64} \) feet per second. Since the object is launched from the ground, the initial height is \( s\mathrm{o} = \mathrm{o} \) feet. Create the mathematical model by substitut... | No |
Given and, find \( \left( {f/g}\right) \left( x\right) \) and determine the restrictions. | In this case, the domain of \( f \) consists of all real numbers except -3 and 7, and the domain of \( g \) consists of all real numbers except 7 and -7 . In addition, the reciprocal of \( g\left( x\right) \) has a restriction of -3 and . Therefore, the domain of this quotient consists of all real numbers except \( - 3... | No |
Example 7\n\nSolve:\n\nSolution:\n\nWhen cross multiplying, be sure to group \( {5n} - 1 \) .\n\n\[ \left( {{5n} - 1}\right) \cdot 2 = 5 \cdot {3n} \]\n\nApply the distributive property in the next step. | Answer:\n\nCross multiplication can be used as an alternate method for solving rational equations. The idea is to simplify each side of the equation to a single algebraic fraction and then cross multiply. | No |
Simplify. | Since the indices are odd, the absolute value is not used.\n\nIn summary, for any real number \( a \) we have,\n\nWhen \( n \) is odd, the \( n \) th root is positive or negative depending on the sign of the radicand.\n\nWhen \( n \) is even, the \( n \) th root is positive or not real depending on the sign of the radi... | No |
Simplify: | Solution:\n\nHere 150 can be written as\n\nWe can verify our answer on a calculator:\n\nAlso, it is worth noting that Answer: | No |
Simplify: | Here we note that the index is odd and the radicand is negative; hence the result will be\n\nnegative. We can factor the radicand as follows:\n\nThen simplify:\n\nAnswer: | No |
Simplify: | Solution: Answer: Try this! Answer: | No |
Multiply: | Solution:\n\nApply the product rule for radicals, and then simplify.\n\nAnswer: | No |
Simplify: | Here the radicand of the square root is a cube root. After rewriting this expression using rational exponents, we will see that the power rule for exponents applies. | No |
Solve: | Solution:\n\nBegin by squaring both sides of the equation.\n\nThe resulting quadratic equation can be solved by factoring.\n\nChecking the solutions after squaring both sides of an equation is not optional. Use the original equation when performing the check.\n\n<table><tr><td>\\( x \\)</td><td>\\( \\checkmark \\)</td>... | Yes |
Solve: | Begin by isolating the radical.\n\nSince we squared both sides, we must check our solutions.\n\n<table><tr><td>\\( x \\)</td><td>\\( x \\)</td></tr></table>\n\nSince both possible solutions are extraneous, the equation has no solution.\n\nAnswer: No solution, \\( \\varnothing \\) | Yes |
Solve: | Solution:\n\nEliminate the radicals by cubing both sides.\n\nCheck.\n\n<table><tr><td>3</td></tr></table>\n\nAnswer: The solutions are \( \pm \mathbf{8} \) . | No |
Add: | Add the real parts and then add the imaginary parts. | No |
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