Complemented characteristic subgroup

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

Definition

Symbol-free definition

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

1. It is a characteristic subgroup as well as a permutably complemented subgroup.
2. It is a characteristic subgroup as well as a lattice-complemented subgroup.
3. It is a characteristic subgroup as well as a complemented normal subgroup, i.e., it occurs as the normal subgroup part in an internal semidirect product.

Relation with other properties

Stronger properties

• Left-transitively complemented normal subgroup
• Characteristically complemented characteristic subgroup
• Characteristic direct factor

Weaker properties

• Characteristic subgroup
• Complemented normal subgroup
  • Permutably complemented subgroup
  • Lattice-complemented subgroup
• Normal subgroup