Group whose center is a direct factor

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group whose center is a direct factor is a group satisfying the following equivalent conditions:

1. It is the direct product of a centerless group and an abelian group.
2. Its center is a permutably complemented subgroup.
3. Its center is a complemented normal subgroup.
4. Its center is a direct factor.

Relation with other properties

Stronger properties

• Abelian group
• Centerless group
• Simple group
• Characteristically simple group
• Group in which every normal subgroup is a direct factor

Weaker properties

• Group whose center is an AEP-subgroup
• Group having an automorphism whose restriction to the center is the inverse map

Opposite properties

• Group whose center is normality-large
• (non-abelian) nilpotent group