Distinct left(right) cosets

[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.

 

[blue_raver22]

Yes, your understanding is correct. Since G is a group and H is a subgroup of G, the left coset aH and the right coset Ha will contain the same elements, but in different orders. Therefore, if aH and bH are distinct left cosets, then Ha and Hb will also be distinct right cosets. This is because the order of multiplication in a group does not affect the elements themselves, but only their arrangement. Therefore, the left and right cosets will be distinct but contain the same elements.