Question in Abstract Algebra?

Question

Let G be a group and H a subgroup of G. Define N(H) = {x is an element of G | xHx^-1 = H}. How can you prove that N(H) is a subgroup of G?

Answer 1

1 is in N(H) because 1^-1 = 1

1H1^-1 = H

and 1 is in G.

If x,y are in N(H) (xy)^-1 = y^1*x^-1

xyH(xy)^-1 = H, thus xy is in N(H)

If x is in N(H), x^-1 is in N(H) because (x^-1)^-1 = x

x^-1Hx = H

If x,y,z are in N(H), because N(H) is a subset of G:
(xy)z = x(yz)

Thus, N(H) is a subgroup of G.