Prove H is a Subgroup of G: ab=ba

[Punkyc7]

If ab=ba in a group G, let H={g$\in$G|agb=bga}.

Show that H is a subgroup.I have the identity because ab=ba implies that aeb=bea$\in$H.

I can't seem to figure out how to get the inverses or closer. Any suggestions or slight nudges in the right direction?

 

[vela]

Let x, y ∈ H. You want to show that xy^-1 ∈ H, so consider a(xy^-1)b.

a(xy^-1)b = ax(bb^-1)y^-1(a^-1a)b = (axb)(b^-1y^-1a^-1)(ab)

Use what you know about a, b, x, and y to show that that product is equal to b(xy^-1)a.

 

[Punkyc7]

Thanks that was clever... didnt think of adding A's and B's in the middle and inversing.

 

[vela]

It's a pretty common technique to create combinations you know how to deal with. Good to have in your toolkit.