Solvability of fixed length is direct product-closed

Statement

Version in terms of fixed class bound

Suppose $G_i,i\in I$ is a collection of groups indexed by an indexing set $I$. Suppose there is a positive integer $\ell$ such that each $G_i$ is a solvable group of derived length at most $\ell$.

Then, the external direct product of the $G_i$s is also a solvable group of derived length at most $\ell$.

Version in terms of maximum class

Suppose $G_i,i\in I$ is a collection of groups indexed by an indexing set $I$. If all the $G_i$s are solvable groups and there is a common finite bound on their derived length values, then the external direct product of the $G_i$s is also a solvable group and its derived length is the maximum of the derived length values of all the $G_i$s.

In particular, for two solvable groups $G_1$ and $G_2$ of derived lengths ${\ell }_1,{\ell }_2$ respectively, the derived length of $G_1\times G_2$ equals $max\{{\ell }_1,{\ell }_2\}$.

Related facts

• Nilpotency of fixed class is direct product-closed
• Derived series is direct product-closed