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