# Thread: Groups - just want to check if i'm on the right track!?

1. ## Groups - just want to check if i'm on the right track!?

H, K sub-groups of a group G. x,y elements of G. Suppose Hx=Ky. Show that x(y^-1) is an element of H, and deduce that H = K.

I just multiplied x^-1 to get H=Ky(x^-1), to get yx^-1 element of H, since H is a sub-group, it's inverse, y^-1x is also an element of H...

can i say that?

How would I show that H = K?

Thanks!

2. Originally Posted by rmangan
H, K sub-groups of a group G. x,y elements of G. Suppose Hx=Ky. Show that x(y^-1) is an element of H, and deduce that H = K.
since $y \in Ky = Hx,$ there exists $h \in H$ such that $y=hx.$ hence $xy^{-1}=h^{-1} \in H.$ thus: $K=Hxy^{-1} =H.$