# Homework Help: Subgroup Question

1. Jan 25, 2012

### 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 cant seem to figure out how to get the inverses or closer. Any suggestions or slight nudges in the right direction?

2. Jan 25, 2012

### vela

Staff Emeritus
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.

3. Jan 25, 2012

### Punkyc7

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

4. Jan 25, 2012

### vela

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