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.

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).

Individual evidence

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