# Group having no proper cocentral subgroup

From Groupprops

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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## 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.