Subgroup, index of a

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in a group

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


[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:,_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