# 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]
This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
View other such subgroup properties

## 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 $N \le H \le G$ 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 Subnormal subgroup of finite index|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 $\mathbb{Z}$ Subgroup of finite index in finitely generated group|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) Subgroup of 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) Automorph-commensurable subgroup, Nearly normal subgroup|FULL LIST, MORE INFO
automorph-commensurable subgroup its intersection with any of its automorphic subgroups has finite index in both |FULL LIST, MORE INFO

## 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.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

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
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

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 $K \le G$ 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 $G$.

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.