G is a group, H - subgroup of G. Let Hx = {hx | h is in H}. Show that, given a and b is in G, then Ha=Hb?

Question

Also show that Ha ∩ Hb = empty set

Answer 1

That is not always true. Ha=Hb if and only if Ha*b^-1 = H. What your statement says is only true sometimes (not always).

Whoops (Correct now)

Answer 2

This is not necessarily true. Simple example is: G is the group of integers with addition, and H is the subgroup of multiples of 2. If a = 1 and b = 2, then

H + a = { all odd numbers }
H + b = { all even numbers }

which are obviously not the same.

Answer 3

This is not always true. In fact, Ha=Hb if and only if b is in Ha, so ba^(-1) is in H.

The previous poster multiplied by b^(-1) on the wrong side.