Lagrange's Theorem

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• Sep 24th 2009, 10:23 AM
Godisgood
Lagrange's Theorem
Let G be a group and H a subgroup of G. Define, for a,b elements of G, a is an equivalence relation to b if a^-1b element of H. Prove that this defines an equivalence relation on G and show that [a] = aH = {ah,h is an element of H}. The sets aH are called left cosets of H in G
• Sep 24th 2009, 12:10 PM
lepton
This is pretty straight forward, just check the three properties of an equivalence relation. Recall an equivalence relation is reflexive, symmetric, and transitive. Here's a start.

Let $a,b,c\in H$

If $ab^{-1}\in H$, then since $H$ is a subgroup, $(ab^{-1})^{-1}=ba^{-1}\in H$ (symmetry)

Now you need only show:

$aa^{-1}\in H$ (reflexivity)

and

If $ab^{-1}\in H$ and $bc^{-1}\in H$, then $ac^{-1}\in H$ (transitivity)

For the second part, show containment both ways, $[a]\subset aH$ and $aH\subset [a]$. It can be done in one fell swoop with a short string of equivalences if implications in both directions seems redundant.
• Sep 24th 2009, 01:46 PM
Godisgood
Thanks Lepton..
I am stil having trouble showing that [a] is a subset of aH and aH is a subset of [a]..Please can you take me through the steps??