Solvability of fixed length is direct product-closed

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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_is 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_is are solvable groups and there is a common finite bound on their derived length values, then the external direct product of the G_is is also a solvable group and its derived length is the maximum of the derived length values of all the G_is.

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 \}.

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