Subgroup, index of a
From Encyclopedia of Mathematics
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.
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=24856
Subgroup, index of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subgroup,_index_of_a&oldid=24856
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article