# Difference between revisions of "Subgroup, index of a"

in a group \$G\$

The number of cosets (cf. Coset in a group) in any decomposition of \$G\$ with respect to this subgroup \$H\$ (in the infinite case, the cardinality of the set of these cosets). If the number of cosets is finite, \$H\$ is called a subgroup of finite index in \$G\$. The intersection of a finite number of subgroups of finite index itself has finite index (Poincaré's theorem). The index of a subgroup \$H\$ in \$G\$ is usually denoted by \$|G:H|\$. The product of the order of a subgroup \$H\$ by its index \$|G:H|\$ is equal to the order of \$G\$ (Lagrange's theorem). This relationship applies to finite groups \$G\$ and also to infinite groups \$G\$ for the corresponding cardinalities.

#### References

 [1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) [2] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
How to Cite This Entry:
Subgroup, index of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subgroup,_index_of_a&oldid=13599
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article