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 $p$-Sylow subgroup of $G$. Now prove that $HP$ is semidirect product of $P$ by $H$. b) Say that a finite group $G$ has the property (ST) if there exists a numbering $p_1, \ldots, p_s$ of the distinct prime numbers dividing the order

of $G$, and $p_i$-Sylow 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 $G$. Now prove 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 $p_1^{a_1} \ldots p_i^{a_i}$; moreover, $G_i$ is semidirect product of $P_i$ by $G_{i-1}$. c) Now prove that a finite group $G$ has the property (ST) if and only if there exists a Sylow subgroup $P$ of $G$, normal in $G$ and such that $G/P$ has the property (ST). d) Now prove that every subgroup, every quotient group and every central extension of a group having the property (ST) has the property (ST).