# Composition factor-equivalent groups

From Groupprops

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

## Definition

Suppose and are groups of finite composition length. We say that and are **composition factor-equivalent** if the multisets of composition factors of and 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 as in the composition series of .

## 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.