Index of a subgroup

Symbol-free definition
The index of a subgroup in a group is the following equivalent things:


 * 1) The number of left cosets of the subgroup
 * 2) The number of right cosets of the subgroup

The collection of left cosets is sometimes termed the coset space, so in this language, the index of a subgroup is the cardinality of its coset space.

Definition with symbols
Given a subgroup $$H$$ of a group $$G$$, the index of $$H$$ in $$G$$, denoted $$[G:H]$$, is defined in the following ways:


 * 1) It is the number of left cosets of $$H$$ in $$G$$, i.e. the number of sets of the form $$xH$$.
 * 2) It is the number of right cosets of $$H$$ in $$G$$, i.e. the number of sets of the form $$Hx$$.

The collection of left cosets of $$H$$ in $$G$$ is sometimes termed the coset space, and is denoted $$G/H$$. With this notation, the index of $$H$$ in $$G$$, is the cardinality $$\left|G/H\right|$$.

Equivalence of definitions
The equivalence of definitions follows from the fact that there is a natural bijection between the collection of left cosets of a subgroup, and the collection of its right cosets, given by the map $$g \mapsto g^{-1}$$

Further note for finite groups
When the group is finite, then by Lagrange's theorem, the index of a subgroup is the ratio of the order of the group to the order of the subgroup.

Multiplicativity of the index
If $$H \le K \le G$$, then we have:

$$[G:K][K:H] = [G:H]$$

In other words, the number of cosets of $$H$$ in $$G$$ equals the number of cosets of $$H$$ in $$K$$, times the number of cosets of $$K$$ in $$G$$.

In fact, more is true. We can set up a bijection as follows:

$$G/K \times K/H \to G/H$$

However, this bijection is not a natural one, and, in order to define it, we first need to choose a system of coset representatives of $$H$$.

Effect of intersection on the index
If $$H_1$$ and $$H_2$$ are two subgroups of $$G$$, then the index of $$H_1 \cap H_2$$ is bounded above by the product of the indices of $$H_1$$ and of $$H_2$$.

This follows as a consequence of the product formula. Note that equality holds if and only if $$H_1H_2 = G$$.

Note that in case $$H_1$$ and $$H_2$$ are conjugate subgroups of index $$r$$, the index of $$H_1 \cap H_2$$ is bounded above by $$r(r-1)$$.

For double cosets and multicosets

 * Double coset index of a subgroup
 * Multicoset index of a subgroup

Related subgroup properties

 * Subgroup of index two
 * Subgroup of finite index
 * Subgroup of prime index
 * Hall subgroup

Textbook references

 * , Page 57, Point (6.8) (definition in paragraph, defined as number of left cosets)
 * , Page 90 (formal definition, defined as number of left cosets)