Exercise 13, § 4

Let $G$ be a group. If $S$ is a simple group, $S$ is said to occur in $G$ if there exist two subgroups $H, H'$ of $G$ with $H$ a normal subgroup of $H'$ such that $H'/H$ is isomorphic to $S$. Let $\operatorname{in}(G)$ denote the set of isomorphism classes of simple groups occurring in $G$. (a) Show that $\operatorname{in}(G) = \varnothing \Leftrightarrow G = \{e\}$. (b) If $H$ is a subgroup of $G$, show that $\operatorname{in}(H) \subset \operatorname{in}(G)$; if moreover $H$ is normal, then

(c) Let $G_1$ and $G_2$ be two groups. Show the equivalence of the two following properties: (i) $\operatorname{in}(G_1) \cap \operatorname{in}(G_2) = \varnothing$; (ii) Every subgroup of $G_1 \times G_2$ is of the form $H_1 \times H_2$ with $H_1 \subset G_1$ and $H_2 \subset G_2$. (Use Exercise 8.) When $G_1$ and $G_2$ are finite groups, show that (i) and (ii) are equivalent to: (iii) Card($G_1$) and Card($G_2$) are relatively prime.