# One-headed group

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

This is a variation of simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

## Contents

## Definition

### Symbol-free definition

A group is said to be **one-headed** if it has a proper normal subgroup that contains every proper normal subgroup. Note that such a proper normal subgroup must also therefore be the unique maximal normal subgroup. The quotient of the group by this maximal normal subgroup is termed the **head** of the group.

Note that simply saying that there is a unique maximal normal subgroup is a somewhat weaker statement, though it is equivalent for a group in which every proper subgroup is contained in a maximal subgroup.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Simple group | nontrivial, no proper nontrivial normal subgroup | |FULL LIST, MORE INFO | ||

Composition series-unique group | has a unique composition series | |FULL LIST, MORE INFO | ||

Quasisimple group | perfect group whose inner automorphism group is a simple group | |FULL LIST, MORE INFO |

### Related properties

- Monolithic group: This has a unique nontrivial normal subgroup contained in all the nontrivial normal subgroups.

### Analogues in other algebraic structures

- Local ring in the theory of commutative unital rings.
- Local ring in the theory of noncommutative unital rings.