# Thread: Abstract Algebra proof - cosets

1. ## Abstract Algebra proof - cosets

Let H be a subgroup of G such that g^-1hg is an element of H for all g in G and all h in H. Show that every left coset gH is the same as the right coset Hg.

need to show gh1=h2g
I know I need to show this, but am unsure on how to.
I know the whole set up excpet for this part and it confuses me.

2. maybe i'm just really bad at math, but... isn't this question in the wrong forum area?? just a though. After all.. it is abstract algebra, and i think MHF has a different forum section for those types of questions.

3. This should be in:

Universtiy Math Help > Number Theory

4. Originally Posted by kathrynmath
Let H be a subgroup of G such that g^{-1}hg is an element of H for all g in G and all h in H. Show that every left coset gH is the same as the right coset Hg.

need to show gh1=h2g
I know I need to show this, but am unsure on how to.
I know the whole set up excpet for this part and it confuses me.

I'm a little rusty on the group theory stuf but I can recall the following:

Every subgroup of an abelian group is normal. As your subgroup H is normal as $g^-1hg \in H$ maybe we can assume G is abelian (help anyone!)

So if you can prove that G is in fact abelian then the following should follow for your cosets

if $H \triangleleft G$ then

$g_1Hg_2H$

operating on cosets

$\Rightarrow g_1g_2H$
$\Rightarrow g_2g_1H$

I hope this helps some.