Complemented isomorph-containing subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: complemented normal subgroup and isomorph-containing subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup of a group is termed a complemented isomorph-containing subgroup if it satisfies the following equivalent conditions:

1. It is both a permutably complemented subgroup and an isomorph-containing subgroup.
2. It is both a lattice-complemented subgroup and an isomorph-containing subgroup.
3. It is both a complemented normal subgroup and an isomorph-containing subgroup.

Relation with other properties

Stronger properties

• Fully invariant direct factor: For full proof, refer: Equivalence of definitions of fully invariant direct factor
• Normal Hall subgroup
• Normal Sylow subgroup
• Complemented homomorph-containing subgroup

Weaker properties

• Isomorph-containing subgroup
• Complemented characteristic subgroup
• Complemented normal subgroup