Exercise 37, § 6

(a) Let $G$ be a finite group, $H$ a normal subgroup of $G$, $p$ a prime number not dividing the order of $H$ and $P$ a Sylow $p$-subgroup of $G$. Show that HP is the semi-direct product of $P$ by $H$.

(b) We say that a finite group $G$ has property (ST) if there exists a numbering $p_1, \ldots, p_s$ of distinct prime numbers dividing the order

of $G$ and Sylow $p_i$-subgroups $P_1, \ldots, P_s$ of $G$ such that, for $1 \leq i \leq s$, the set $G_i = P_1 P_2 \ldots P_i$ is a normal subgroup of $G$. Show that, if this is so, $G_i$ does not depend on the choice of the $P_i$; it is the unique subgroup of $G$ of order $p_1^{a_1} \cdots p_i^{a_i}$; further, $G_i$ is the semi-direct product of $P_i$ by $G_{i-1}$.

(c) Show that a finite group $G$ has property (ST) if and only if there exists a Sylow subgroup $P$ of $G$ which is normal in $G$ and such that $G/P$ has property (ST).

(d) Show that every subgroup, every quotient group and every central extension of a group with property (ST) has property (ST).