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Groupprops β

Group in which every subgroup of finite index has finitely many automorphic subgroups

Definition

Equivalent definition in tabular format

No. Shorthand A group G is a group in which every subgroup of finite index has finitely many automorphic subgroups if ...
1 finitely many automorphic subgroups for every subgroup of finite index H in G, there are only finitely many automorphic subgroups to H in G, i.e., there are only finitely many subgroups K of G for which there exists an automorphism \sigma of G such that \sigma(H) = K.
2 characteristic core has finite index for every subgroup of finite index H in G, the characteristic core of H in G has finite index in G.
3 contains characteristic subgroup of finite index for every subgroup of finite index H in G, there exists a characteristic subgroup of finite index L in G such that L is contained in H.
4 normal subgroup, finitely many automorphic subgroups for every normal subgroup of finite index H in G, there are only finitely many automorphic subgroups to H in G, i.e., there are only finitely many subgroups K of G for which there exists an automorphism \sigma of G such that \sigma(H) = K.
5 normal subgroup, characteristic core of finite index for every normal subgroup of finite index H in G, the characteristic core of H in G has finite index in G.
6 normal subgroup contains characteristic subgroup of finite index for every normal subgroup of finite index H in G, there exists a characteristic subgroup of finite index L in G such that L is contained in H.

Relation with other properties