Normal subgroup of finite index: Difference between revisions
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A '''normal subgroup of finite index''' in a group is a subgroup satisfying the following equivalent conditions: | A '''normal subgroup of finite index''' in a group is a subgroup satisfying the following equivalent conditions: | ||
# It is [[normal subgroup|normal]] and its [[index]] in the whole group is finite | |||
# It is the kernel of a [[homomorphism]] to a [[finite group]] | |||
# It is the [[normal core]] of a [[subgroup of finite index]] | |||
===Equivalence of definitions=== | |||
The equivalence of definitions (1) and (2) follows from the [[first isomorphism theorem]]. The equivalence with definition (3) follows from [[Poincare's theorem]]. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 14:21, 26 March 2009
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup of finite index
View other subgroup property conjunctions | view all subgroup properties
Definition
Symbol-free definition
A normal subgroup of finite index in a group is a subgroup satisfying the following equivalent conditions:
- It is normal and its index in the whole group is finite
- It is the kernel of a homomorphism to a finite group
- It is the normal core of a subgroup of finite index
Equivalence of definitions
The equivalence of definitions (1) and (2) follows from the first isomorphism theorem. The equivalence with definition (3) follows from Poincare's theorem.