virtually abelian subgroup theorem

Let us suppose that G is virtually abelian and H is an abelianMathworldPlanetmath subgroupMathworldPlanetmathPlanetmath of G with a the finite right cosetMathworldPlanetmath partitionMathworldPlanetmath

G=HeHx2Hxq, *

so if K is any other subgroup in G we are going to prove:

K is also virtually abelian

Proof: From (*) above we have

=(KH)(KHx2)(KHxq). **

Here we consider the two cases:
1) xiK
2) xjK
In the first case K=Kxi, and then KHxi=KxiHxi=(KH)xi. In the second, find yjKHxj hence KHxj=KyjHyj=(KH)yj
So, in the equation (**) above we can replace (reordering subindexation perhaps) to get


relation which shows that the index [K:KH][G:H].
It could be < since it is posible that KHxr= for some indexes r

Title virtually abelian subgroup theorem
Canonical name VirtuallyAbelianSubgroupTheorem
Date of creation 2013-03-22 18:58:42
Last modified on 2013-03-22 18:58:42
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 13
Author juanman (12619)
Entry type Theorem
Classification msc 20F99
Classification msc 20E99
Classification msc 20E07
Synonym subgroup theorem