Just infinite group

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Symbol-free definition

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

  1. It is infinite and every nontrivial normal subgroup is of finite index, i.e., is a normal subgroup of finite index.
  2. It is infinite and every proper quotient is finite.
  3. 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.
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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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  • 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