Prove H Contains in gH ≠ g-H: Counting Principles

[Goomany]

Homework Statement

Suppose that H is a subgroup of G such that whenever H a is not equal to H b.
Then a H not equal to b H.
Prove that g H g ^-H
Contains H.
For all g Is an element of G.[/B]

Homework Equations

The Attempt at a Solution

I tried the contrapositive position
( sorry, I'm doing this on an iPhone and I can't access latex )

 

[fresh_42]

Are you sure you haven't a mistake in there? $H \subseteq gHg^{-1}H$ for all $g\in G$ means, $H \trianglelefteq G$ is a normal subgroup. What about $G=Sym (3) = \{1,(12),(13),(23),(123),(132)\}$ and $H=\{1,(12)\}$?