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Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups

Statement

Verbal statement

Any finitely generated group is a group in which every subgroup of finite index has finitely many automorphic subgroups.

Statement with symbols

The statement has the following equivalent formulation:

  1. Suppose H is a subgroup of finite index in a finitely generated group G. Then, there are only finitely many subgroups K of G that are automorphic subgroups of H, i.e., for which there exists an automorphism \sigma satisfying \sigma(H) = K.
  2. Suppose H is a subgroup of finite index in a finitely generated group G. Then, the characteristic core of H in G is also a subgroup of finite index in G.
  3. Suppose H is a subgroup of finite index in a finitely generated group G. Then, H contains a characteristic subgroup of finite index in G.

Related facts

Facts used