View source for Subgroup, index of a

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{{MSC|20A05}}

''in a group $G$''

The number of cosets (cf. [[Coset in a group|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====
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|valign="top"|{{Ref|KaMeMe}}||valign="top"|  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups", Springer  (1979)  (Translated from Russian)
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|valign="top"|{{Ref|Ku}}||valign="top"|  A.G. Kurosh,   "The theory of groups", '''1–2''', Chelsea  (1955–1956)  (Translated from Russian)
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