Complemented homomorph-containing subgroup
From Groupprops
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
Contents
Definition
A subgroup of a group is termed a complemented homomorph-containing subgroup if it satisfies the following equivalent conditions:
- It is both a complemented normal subgroup and a homomorph-containing subgroup of the whole group.
- It is both a complemented normal subgroup and a normal subgroup having no nontrivial homomorphism to its quotient group.
- It is both a permutably complemented subgroup and a homomorph-containing subgroup of the whole group.
- It is both a lattice-complemented subgroup and a homomorph-containing subgroup of the whole group.