# 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 . ${\ displaystyle G}$${\ displaystyle U}$${\ displaystyle G / U}$${\ displaystyle U \ backslash G}$${\ displaystyle U}$${\ displaystyle G}$${\ displaystyle (G \ colon U)}$${\ displaystyle [G \ colon U]}$${\ displaystyle | G \ colon U |}$

## properties

• It applies . (Here denotes the order of .)${\ displaystyle (G \ colon 1) = | G |}$${\ displaystyle | G |}$${\ displaystyle G}$
• The index is multiplicative, i. H. is a subset of , and a subset of , the following applies ${\ displaystyle U}$${\ displaystyle G}$${\ displaystyle V}$${\ displaystyle U}$
${\ displaystyle (G \ colon V) = (G \ colon U) \ cdot (U \ colon V).}$
• The special case is often referred to as Lagrange's theorem (after J.-L. Lagrange ): ${\ displaystyle V = 1}$
The following applies to a group and a subgroup : ${\ displaystyle G}$${\ displaystyle U}$
${\ displaystyle | G | = (G \ colon U) \ cdot | U |.}$
In the case of finite groups, the index of a subgroup can be written as
${\ displaystyle (G \ colon U) = {\ frac {| G |} {| U |}}}$
to calculate.
• If a normal divisor , then the index of in is just the order of the factor group , that is ${\ displaystyle N \ vartriangleleft G}$${\ displaystyle N}$${\ displaystyle G}$ ${\ displaystyle G / N}$
${\ displaystyle (G \ colon N) = \ left | G / N \ right |}$.
• 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 .${\ displaystyle U}$${\ displaystyle G}$${\ displaystyle p> 1}$${\ displaystyle | G |}$${\ displaystyle U}$${\ displaystyle G}$

## 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.

• 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).

## Individual evidence

1. Hungerford (1989), p. 89
2. Hungerford (1989), p. 247