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 $\left|G:H\right|$. 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) |
Subgroup, index of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subgroup,_index_of_a&oldid=24868