Composition factor-equivalent groups

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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.