Difference between revisions of "Subgroup, index of a"
From Encyclopedia of Mathematics
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| − | 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 $|G:H|$. The product of the order of a subgroup $H$ by its index $|G:H|$ 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. | + | 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. |
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====References==== | ====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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| + | |} | ||
Revision as of 11:14, 20 April 2012
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
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