Normal subgroup of finite index
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.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| characteristic subgroup of finite index | characteristic subgroup and its index in the whole group is finite. | characteristic implies normal | any finite example for normal not implies characteristic | |FULL LIST, MORE INFO |
| normal subgroup of finite group | normal subgroup and the whole group is a finite group. | normal subgroup of finite index|normal subgroup of finite group}} |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| subgroup of finite index | |FULL LIST, MORE INFO | |||
| normal subgroup with finitely generated quotient group | |FULL LIST, MORE INFO | |||
| quotient-powering-invariant subgroup | normal subgroup of finite index implies quotient-powering-invariant | take any finite normal subgroup in an infinite group; see finite normal implies quotient-powering-invariant | |FULL LIST, MORE INFO | |
| powering-invariant subgroup | |FULL LIST, MORE INFO |