Subgroup of finite index

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
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Definition

Equivalent definitions in tabular format

No.	Shorthand	A subgroup of a group is termed a subgroup of finite index if ...	A subgroup $H$ of a group $G$ is termed a subgroup of finite index if ...
1	finite index	its index in the whole group is finite.	the index $|G:H|$, defined as the size of the coset space $[G:H]$, is finite.
2	contains normal subgroup of finite index	it contains a normal subgroup of finite index.	there is a normal subgroup $N$ of $G$ such that $N\leq H\leq G$ and the quotient group $G/N$ is finite.
3	normal core has finite index	its normal core in the whole group has finite index in the whole group.	the normal core of $H$ in $G$ is a normal subgroup of finite index in $G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subgroup of index two	the index is two; i.e., it has just two cosets	(obvious)	take the trivial subgroup in any finite group of order greater than two	|FULL LIST, MORE INFO
subgroup of prime index	the index is a prime number	(obvious)	take the trivial subgroup in any finite group of non-prime order	|FULL LIST, MORE INFO
normal subgroup of finite index	normal subgroup whose quotient group is finite	(obvious)	take any non-normal subgroup of a finite group	|FULL LIST, MORE INFO
subgroup of finite group	the whole group is finite	(obvious)	any infinite group as subgroup of itself; $n\mathbb {Z}$ as subgroup of $\mathbb {Z}$	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subgroup of finite double coset index	has finitely many double cosets in the whole group	(obvious, since double cosets are unions of cosets)	counterexamples can be found using Borel subgroups in general linear groups	|FULL LIST, MORE INFO
subgroup contained in finitely many intermediate subgroups	the number of intermediate subgroups between the subgroup and whole group is finite	(via finite double coset index)	(via finite double coset index)	|FULL LIST, MORE INFO
almost normal subgroup	its normalizer in the whole group has finite index	(obvious)	trivial subgroup in an infinite group	|FULL LIST, MORE INFO
nearly normal subgroup	it has finite index in its normal closure	(obvious)	trivial subgroup in an infinite group	|FULL LIST, MORE INFO
conjugate-commensurable subgroup	its intersection with any of its conjugate subgroups has finite index in both	(via nearly normal)	(via nearly normal)	|FULL LIST, MORE INFO
automorph-commensurable subgroup	its intersection with any of its automorphic subgroups has finite index in both			|FULL LIST, MORE INFO

Conjunction with other properties

• normal subgroup of finite index

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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The property of having finite index is transitive, viz a subgroup of finite index in a subgroup of finite index again has finite index. This follows essentially from the fact that the index is multiplicative. For full proof, refer: Index is multiplicative

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
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If $H$ has finite index in $G$ and $K$ is any intermediate subgroup, the index of $H$ in $K$ is a factor of the index of $H$ in $G$.

Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition

If $H$ has finite index in $G$ and $K\leq G$ is any subgroup, $H\cap K$ has finite index in $K$. In fact, $[K:H\cap K]$ is bounded from above by $[G:H]$. For full proof, refer: Index satisfies transfer inequality

Intersection-closedness

This subgroup property is finite-intersection-closed; a finite (nonempty) intersection of subgroups with this property, also has this property
View a complete list of finite-intersection-closed subgroup properties

A finite intersection of subgroups of finite index again has finite index. In fact, the index of the intersection is bounded from above by the product of the indices of each subgroup. For full proof, refer: Index satisfies intersection inequality

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties

Any subgroup containing a subgroup of finite index is also of finite index. In fact, if $H$ has finite index in $G$ and $K$ is any intermediate subgroup, the index of $K$ in $G$ is a divisor of the index of $H$ in $G$.

Template:Finite-dirprodclosedsgp

If $H_{1}$ has finite index in $G_{1}$ and $H_{2}$ has finite index in $G_{2}$ then $H_{1}$ × $H_{2}$ has finite index in $G_{1}$ × $G_{2}$ when viewed naturally as a subgroup. In fact, the index is the product of the individual indices.

Facts

As a property operator on groups

The virtually operator on group properties takes as input a group property and gives as output the property of being a group that has a subgroup of finite index satisfying that property.

In relation with generating sets

It turns out that if $H$ has finite index in $G$, $G$ is finitely generated if and only if $H$ is. Moreover, there is a generating set for $H$ whose size is bounded above by the size of the generating set for $G$ times the index of $H$. Similarly, given any generating set for $H$, there is a generating set for $G$ whose size is bounded above by the size of the generating set for $H$ times the logarithm of the index of $H$ in $G$.

The bound on size of generating set of subgroup in terms of that of group comes from a constructive result called Schreier's lemma.

Normal core

The normal core of a subgroup of index $n$ has index at most $n!$. Hence, any subgroup of finite index contains a normal subgroup of finite index. This result is sometimes termed Poincare's theorem.