Exercise 39, § 6

Let G be a finite group, $n = \mathrm{Card}(G)$, $p$ a prime number, X the set of Sylow $p$-subgroups of G and $r = \mathrm{Card}(X)$. Then G operates transitively on X by conjugation; let $\phi : G \to \mathfrak{S}_X \approx \mathfrak{S}_r$ be the corresponding homomorphism (A $\approx$ B denotes the relation "A is isomorphic to B") and K its kernel.

(a) Show that K is the intersection of the normalizers of the $P \in X$ and that the Sylow $p$-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) Show that if G has no subgroup of index 2, then $\phi(G) \subset \mathfrak{A}_X$ and that if $n = kr!$ (resp. $\frac{1}{2}kr!$), then $\phi(G) = \mathfrak{S}_X$ (resp. $\mathfrak{A}_X$).

(c) Using (a), (b) and the preceding exercise, show the following facts: If $n = 12$ and a Sylow 3-subgroup of G is not normal, then $G \approx \mathfrak{A}_4$. If $n = 24$ and G does not have property (ST), then $G \approx \mathfrak{S}_4$. If $n = 36$, then G has property (ST). (Show that if a Sylow 3-subgroup of G is not normal, 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 G does not have property (ST), then G contains a normal subgroup K of order 2 such that $G/K \approx \mathfrak{S}_4$. If $n = 60$ and G does not have property (ST), then $G \approx \mathfrak{A}_5$. (Show that such a group G has no normal 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, arguing as in Exercise 25 of § 5, that $G \approx \mathfrak{A}_5$.)

If $n = 72$ and a Sylow 3-subgroup of $G$ is not normal, then $G$ contains a normal subgroup $K$ such that $G/K$ is isomorphic either to $\mathfrak{S}_4$ or to $\mathfrak{A}_4$.

If $n = 96$ and a Sylow 2-subgroup of $G$ is not normal, then $G$ contains a normal subgroup $K$ such that $G/K \approx \mathfrak{S}_3$.

(d) Show that a group $G$ of order $\leq 100$ is solvable unless $G \approx \mathfrak{A}_5$.