View source for Subgroup, index of a

''in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090870/s0908701.png" />''

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====
<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>