Index (group theory)

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


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 .


  • 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.
  • 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.


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