Subgroup of finite index

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.

Stronger properties

 * Weaker than::Subgroup of index two
 * Weaker than::Subgroup of prime index
 * Weaker than::Normal subgroup of finite index

Weaker properties

 * Stronger than::Subgroup of finite double coset index
 * Stronger than::Subgroup contained in finitely many intermediate subgroups
 * Stronger than::Almost normal subgroup
 * Stronger than::Nearly normal subgroup

Conjunction with other properties

 * Normal subgroup of finite index

Metaproperties
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.

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 $$G$$.

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 $$[G:H]$$.

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.

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 $$H_1$$ &times; $$H_2$$ has finite index in $$G_1$$ &times; $$G_2$$ when viewed naturally as a subgroup. In fact, the index is the product of the individual indices.

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.