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

From Groupprops
Jump to: navigation, search
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finitely generated group. That is, it states that in a Finitely generated group (?), every subgroup satisfying the first subgroup property (i.e., Subgroup of finite index (?)) must also satisfy the second subgroup property (i.e., Subgroup having finitely many automorphic subgroups (?)). In other words, every subgroup of finite index of finitely generated group is a subgroup having finitely many automorphic subgroups of finitely generated group.
View all subgroup property implications in finitely generated groups | View all subgroup property non-implications in finitely generated groups | View all subgroup property implications | View all subgroup property non-implications
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated group) must also satisfy the second group property (i.e., group in which every subgroup of finite index has finitely many automorphic subgroups)
View all group property implications | View all group property non-implications
Get more facts about finitely generated group|Get more facts about group in which 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

Corollaries

Facts used

  1. Finitely generated implies finitely many homomorphisms to any finite group
  2. Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups