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''in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908701.png" />''
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''in a group $G$''
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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 $\left|G:H\right|$. 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.
  
The number of cosets (cf. [[Coset in a group|Coset in a group]]) in any decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908702.png" /> with respect to this subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908703.png" /> (in the infinite case, the cardinality of the set of these cosets). If the number of cosets is finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908704.png" /> is called a subgroup of finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908705.png" />. The intersection of a finite number of subgroups of finite index itself has finite index (Poincaré's theorem). The index of a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908706.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908707.png" /> is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908708.png" />. The product of the order of a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908709.png" /> by its index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s09087010.png" /> is equal to the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s09087011.png" /> (Lagrange's theorem). This relationship applies to finite groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s09087012.png" /> and also to infinite groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s09087013.png" /> for the corresponding cardinalities.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.I. Kargapolov,  J.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR></table>
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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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Latest revision as of 11:18, 20 April 2012

2010 Mathematics Subject Classification: Primary: 20A05 [MSN][ZBL]

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=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