# Group having no proper cocentral subgroup

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 is termed a group having no proper cocentral subgroup if the only cocentral subgroup of the group is the whole group. Here, a cocentral subgroup is defined as a subgroup whose product with the center of the group is the whole group.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Centerless group center is the trivial group (obvious) |FULL LIST, MORE INFO
Non-abelian group in which every proper normal subgroup is central non-abelian, and every proper normal subgroup is contained in the center (obvious)
Non-abelian group whose center equals its socle non-abelian, and every nontrivial normal subgroup contains the center (obvious)
Non-abelian group whose center is comparable with all normal subgroups non-abelian, and every normal subgroup either contains or is contained in the center (obvious)
Maximal class group
Extraspecial group
Quasisimple group nontrivial perfect group whose inner automorphism group is simple (via every proper normal subgroup is central) |FULL LIST, MORE INFO

### Opposite properties

• Abelian group: The only abelian group satisfying this property is the trivial group. Note that for many of the properties stronger than this property, we have to explicitly include non-abelianness as a clause.