# Subgroup, index of a

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2010 Mathematics Subject Classification: Primary: 20A05 [MSN][ZBL]

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 $\left|G:H\right|$. The product of the order of a subgroup $H$ by its index $\left|G:H\right|$ 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

 [KaMeMe] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups", Springer (1979) (Translated from Russian) [Ku] 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=24868
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article