# Lagrange's Theorem

• Sep 24th 2009, 09:23 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, 11:10 AM
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 $\displaystyle a,b,c\in H$

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

Now you need only show:

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

and

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

For the second part, show containment both ways, $\displaystyle [a]\subset aH$ and $\displaystyle aH\subset [a]$. It can be done in one fell swoop with a short string of equivalences if implications in both directions seems redundant.
• Sep 24th 2009, 12:46 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??