Are Distinct Left Cosets and Right Cosets in a Group Related?

[ivyhawk]

G group, H subgroup of G.
Suppose aH and bH are distinct leftcosets then Ha and Hb must be distinct right cosets?


My humble thoughts:
the left coset aH consists of a times everything in H;
Ha consists of everything in H times a.
Then this argument above is true?

 

[Fredrik]

That's not an argument, it's just the definition of the cosets.

 

[ivyhawk]

That's not an argument, it's just the definition of the cosets.

My doubt is whether there is a counter-example such that when aH and bH are distinct left cosets, Ha and Hb are not distinct right cosets, because this statement looks suspicious.

 

[Fredrik]

When H is a normal subgroup, it's very easy to prove that the statement is true, but I don't see why it must be true when H isn't a normal subgroup. So if you're looking for counterexamples, start by thinking of examples of subgroups that aren't normal.