# Just infinite group

From Groupprops

## Definition

### Symbol-free definition

A group is said to be **just infinite** if it satisfies the following equivalent conditions:

- It is infinite and every nontrivial normal subgroup is of finite index, i.e., is a normal subgroup of finite index.
- It is infinite and every proper quotient is finite.
- It is infinite and every subgroup of infinite index is a core-free subgroup, i.e., the normal core of any subgroup of infinite index is trivial.

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Just infinite group, all facts related to Just infinite group) |Survey articles about this | Survey articles about definitions built on this

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View a list of other standard non-basic definitions

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

## Examples

- The group of integers is a just infinite group.
- Any infinite simple group is a just infinite group.

## Relation with other properties

### Stronger properties

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

Infinite simple group | Infinite group that is also a simple group |