# Index (group theory)

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

## definition

Let it be a group and a subgroup. Then the set of left secondary classes and the set of right secondary classes are equal. Their thickness is the index of in and is sometimes referred to as or .

## properties

- It applies . (Here denotes the order of .)
- The index is multiplicative, i. H. is a subset of , and a subset of , the following applies
- The special case is often referred to as Lagrange's theorem (after J.-L. Lagrange ):
- The following applies to a group and a subgroup :
- In the case of finite groups, the index of a subgroup can be written as
- to calculate.

- The following applies to a group and a subgroup :
- If a normal divisor , then the index of in is just the order of the factor group , that is
- .

- A subgroup of index 2 is a normal divisor, because of the two (left) secondary classes, one is the subgroup itself and the other is its complement.
- More general: If is a subgroup of and its index, which is also the smallest divisor of the order , then in is a normal divisor .

## Topological groups

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

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

## See also

- The index of the centralizer of a group element corresponds to the power of its conjugation class.
- In Galois theory , the Galois correspondence gives a connection between the relative indices of subgroups of the Galois group and the relative degrees of body extensions.

## literature

Index in group theory:

- Thomas W. Hungerford: Algebra . 5th edition. Springer, New York 1989, ISBN 0-387-90518-9 , pp. 38 ff .

In topological groups:

- Lev Pontryagin : Topological Groups . Teubner, Leipzig 1957 (Russian: Nepreryvnye gruppy . Translated by Viktor Ziegler).