Composition length of direct product is sum of composition lengths

For two groups
Suppose $$G_1$$ and $$G_2$$ are groups with composition lengths $$a_1$$ and $$a_2$$. Then, the composition length of the external direct product $$G_1 \times G_2$$ is the sum $$a_1 + a_2$$.

In particular, if both $$G_1$$ and $$G_2$$ are groups of finite composition length, then so is $$G_1 \times G_2$$. Conversely, if $$G_1 \times G_2$$ is a group of finite composition length, then so are $$G_1$$ and $$G_2$$.

For multiple groups
Suppose $$G_1,G_2,\dots,G_n$$ are groups with composition lengths $$a_1,a_2,\dots,a_n$$. Then, the composition length of the external direct product $$G_1 \times G_2 \times \dots \times G_n$$ is the sum $$a_1 + a_2 + \dots + a_n$$.

Related facts

 * Composition series of direct product is obtained by piecing together composition series of direct factors
 * Chief length of direct product is sum of chief lengths
 * Composition length of extension group is sum of composition lengths
 * Chief length of extension group is bounded by sum of chief lengths