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