Subgroup of finite index

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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

Symbol-free definition

A subgroup of a group is said to be of finite index if its index in the whole group is finite, or equivalently, if it has only finitely many cosets.

Definition with symbols

A subgroup H of a group G is said to be of finite index if [G:H] is finite.

Relation with other properties

Stronger properties

Weaker properties

Conjunction with other properties

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.


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.