Just infinite group: Difference between revisions
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==Definition== | ==Definition== | ||
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A [[group]] is said to be '''just infinite''' if it satisfies the following equivalent conditions: | 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 [[defining ingredient::subgroup of finite index|finite index]], i.e., is a [[defining ingredient::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 [[defining ingredient::core-free subgroup]], i.e., the [[normal core]] of any subgroup of infinite index is trivial. | |||
{{stdnonbasicdef}} | |||
{{group property}} | |||
==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=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| Infinite simple group || Infinite group that is also a [[simple group]] || || || | |||
|} | |||
Latest revision as of 01:46, 19 May 2010
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: (facts closely related 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 properties
VIEW 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 |