Index of a subgroup

In the mathematical subfield of group theory, the index of a subgroup is a measure of the relative size of the entire group.

Definition

It is a group, and a subgroup. Then the set of left cosets and the set of right cosets are equally powerful. Their thickness is in the index of and with, or sometimes referred to.

Properties

• It is true. ( Here denotes the order of. )

• The index is multiplicative, that is, is a subgroup of and a subset of, the following applies

• The special case is often referred to as a set of Lagrange (after JL Lagrange ):

• A subset of the index 2 is a normal divider, as the two (left ) cosets of a sub- group itself, and the other its complement.

• General: Is a subset of, and its index, which is also the smallest divisor of the order, then a normal subgroup in.

Topological groups

In the context of topological groups subgroups of finite index play a special role:

• A subgroup of finite index if and only open when it is completed. (Open subsets are always closed. )
• Every open subset of a compact group has finite index.