Cosets and Normal Sub-group: Relation between the two concepts
We know that for any group, G, a subgroup H is normal-subgroup
IFF every right coset of H is also it's left coset i.e. Ha = aH for all 'a' in G
What if there is a group where every right coset is a left coset but with some other element
i.e. Ha = bH where a=/=b (for all the right cosets of H, obviously there is one exception He = eH)
Basically what I am saying is that there is a one-to-one mapping between set of right cosets and set of left cosets.
I am not sure if such a structure in groups is relevant / important / logical.
What does the above tell the relation between bH and aH? Are they same?