Hello,

I am trying to prove, or disprove by means of a counterexample the two statements.

For a subgroup H of a group G, with elements a,b $\displaystyle \in$ G, and aH is the coset of H containing a, defined as $\displaystyle aH = \{ah\ : h \in H\}$

(1)

$\displaystyle aH=bH$ implies $\displaystyle Ha^{-1} = Hb^{-1}$

(2)

$\displaystyle aH = bH$ implies $\displaystyle Ha = Hb$

I'm not sure where to start on these.

Thanks!