Exercise 8, § 4

Algebra, I, 133

Let $A$ and $A'$ be two groups and $G$ a subgroup of $A \times A'$ . We write:

N = G \cap (A \times \{e\}), \qquad H = \mathrm{pr}_1(G)

N' = G \cap (\{e\} \times A'), \qquad H' = \mathrm{pr}_2(G).

(a) Show that $N$ is normal in $H$ and $N'$ is normal in $H'$ ; define isomorphisms

H/N \to G/(N \times N') \to H'/N'.

(b) Suppose that $H = A, H' = A'$ , that the groups $A$ and $A'$ are of finite length and that no quotient of a Jordan-Hölder series of $A$ is isomorphic to a quotient of a Jordan-Hölder series of $A'$ . Show that $G = A \times A'$ .

Solution

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