
Cosets
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!

1.
$\displaystyle aH = bH \iff \{ah  h \in H\} = \{bh  h \in H\} \iff \{(ah)^{1}  h \in H\} = \{(bh)^{1} h \in H\}$
And you should be able to see with further manipulations that the implication is indeed true.
2.
Look at the group $\displaystyle S_3 $. Can you find 2 elements that show this implication is not true?

Just to confirm, in (1), the next step would be
$\displaystyle \{(ah)^{1}  h \in H\} = \{(bh)^{1} h \in H\}$
then
$\displaystyle \{(h)^{1}(a)^{1}  h \in H\} = \{(h)^{1}(b)^{1} h \in H\}$
by properties of inverse, then
$\displaystyle \{(h)(a)^{1}  h \in H\} = \{(h)(b)^{1} h \in H\}$
Since if $\displaystyle h \in H, \rightarrow h^{1} \in H$
Then we have the desired result!
Thanks!!