Direct Product of Solvable Groups is Solvable

Theorem

Let $G$ and $H$ be groups which are solvable.

Then their (external) direct product $G \times H$ is also solvable.

Proof

By Image of Canonical Injection is Normal Subgroup, $G \times \set {e_H}$ is a normal subgroup of $G \times H$.

Also, by Quotient Group of Direct Products, $\paren {G \times H} / \paren {G \times \set{e_H} }$ is isomorphic to $H$.

The result then follows from Group is Solvable iff Normal Subgroup and Quotient are Solvable.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 75 \delta$