Automorph-join-closed subnormal subgroup

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Definition

A subgroup of a group is termed automorph-join-closed subnormal if the join of any collection of subgroups that are automorphic to it (i.e., its images under automorphisms of the whole group) is a subnormal subgroup.

Relation with other properties

Stronger properties

• Normal subgroup
• 2-subnormal subgroup
• Subnormal subgroup of finite index

Weaker properties

• Conjugate-join-closed subnormal subgroup
• Finite-automorph-join-closed subnormal subgroup
• Finite-conjugate-join-closed subnormal subgroup

Facts

• Any automorph-join-closed subnormal subgroup of a normal subgroup is conjugate-join-closed subnormal. For full proof, refer: Automorph-join-closed subnormal of normal implies conjugate-join-closed subnormal