Exercise 39, § 6

Let G be a finite group, $n = \mathrm{Card}(G)$, $p$ a prime number, X the set of the $p$-Sylow subgroups of G, and $r = \mathrm{Card}(X)$. Then G operates transitively on X by conjugation; let $\varphi : G \to \mathfrak{S}_X \approx \mathfrak{S}_r$, be the corresponding homomorphism (we write A $\approx$ B for the relation “A is isomorphic to B”), and let K be its kernel. a) Prove that K is the intersection of the normalizers of the P $\in$ X, and that the $p$-Sylow subgroup of K is the intersection of the P $\in$ X. Deduce that if $r > 1$ the order k of K divides $n/rp$. b) Prove that if G has no subgroup of index 2, then $\varphi(G) \subset \mathfrak{A}_X$, and that if $n = k\ r!$ (resp. $\frac{1}{2}k\ r!$), then $\varphi(G) = \mathfrak{S}_X$ (resp. $\mathfrak{A}_X$). c) Using a) and b), as well as the preceding exercise, prove the following facts: If $n = 12$, and if a 3-Sylow subgroup of G is not distinguished, then G $\approx \mathfrak{A}_4$. If $n = 24$, and if G does not have the property (ST), then G $\approx \mathfrak{S}_4$. If $n = 36$, then G has the property (ST). (Prove that if a 3-Sylow subgroup of G is not distinguished, then G contains a subgroup K of order 3, such that G/K $\approx \mathfrak{A}_4$, and that K is central.) If $n = 48$, and if G does not have the property (ST), then G contains a distinguished subgroup K of order 2 such that G/K $\approx \mathfrak{S}_4$. If $n = 60$, and if G does not have the property (ST), then G $\approx \mathfrak{A}_5$. (Prove that such a group G has no distinguished subgroup of order 5, nor of order 2, nor of index 2. Deduce that G is isomorphic to a subgroup of $\mathfrak{S}_6$, and by reasoning as in exerc. 25 of I, p. 133, that G $\approx \mathfrak{A}_5$.) If $n = 72$, and if a 3-Sylow subgroup of G is not distinguished, then G contains a distinguished subgroup K such that G/K is isomorphic either to $\mathfrak{S}_4$, or to $\mathfrak{A}_4$. If $n = 96$, and if a 2-Sylow subgroup of G is not distinguished, then G contains a distinguished subgroup K such that G/K $\approx \mathfrak{S}_3$. d) Prove that a group G of order $\leq 100$ is solvable except if G $\approx \mathfrak{A}_5$.