Composition factor-equivalent groups

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

Suppose $G$ and $H$ are groups of finite composition length. We say that $G$ and $H$ are composition factor-equivalent if the multisets of composition factors of $G$ and $H$ are the same (possibly, occurring in a different order). In other words, every simple group has the same number of isomorphic copies in the composition series of $G$ as in the composition series of $H$.

Facts

• The trivial group is not composition factor-equivalent to any other group.
• A simple group is not composition factor-equivalent to any other group.
• Any two finite groups that are composition factor-equivalent must have the same order.
• Two finite solvable groups are composition factor-equivalent if and only if they have the same order.