Complemented homomorph-containing subgroup

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

Definition

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

1. It is both a complemented normal subgroup and a homomorph-containing subgroup of the whole group.
2. It is both a complemented normal subgroup and a normal subgroup having no nontrivial homomorphism to its quotient group.
3. It is both a permutably complemented subgroup and a homomorph-containing subgroup of the whole group.
4. It is both a lattice-complemented subgroup and a homomorph-containing subgroup of the whole group.

Relation with other properties

Stronger properties

• Normal Sylow subgroup
• Normal Hall subgroup
• Fully invariant direct factor

Weaker properties

• Complemented fully invariant subgroup
• Complemented isomorph-containing subgroup
• Complemented characteristic subgroup
• Complemented normal subgroup
• Homomorph-containing subgroup
  • Fully invariant subgroup
  • Isomorph-containing subgroup
  • Characteristic subgroup
  • Normal subgroup