estimation of index of intersection subgroup

Theorem.  If H1,H2,,Hn are subgroupsMathworldPlanetmathPlanetmath of G, then


Proof.  We prove here only the case  n=2;  the general case may be handled by the inductionMathworldPlanetmath.

Let  H1H2:=K.  Let R be the set of the right cosetsMathworldPlanetmath of K and Ri the set of the right cosets of Hi  (i=1, 2).  Define the relationMathworldPlanetmathPlanetmath ϱ from R to R1×R2 as


We then have the equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( conditions


whence ϱ is a mapping and injectivePlanetmathPlanetmath,  ϱ:RR1×R2.  i.e. it is a bijection from R onto the subset  {ϱ(x)xR}  of R1×R2.  Therefore,


As a consequence one obtains the

Theorem (Poincaré).  The index of the intersectionMathworldPlanetmath of finitely many subgroups with finite indices ( is finite.

Title estimation of index of intersection subgroup
Canonical name EstimationOfIndexOfIntersectionSubgroup
Date of creation 2013-03-22 18:56:46
Last modified on 2013-03-22 18:56:46
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Theorem
Classification msc 20D99
Synonym index of intersection subgroup
Related topic LogicalAnd
Related topic Cardinality