MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## coincidence of left/right cosets and normal subgroup

In brief: suppose that the left and right cosets coincides as two coverings of the group as a set, does it imply that the subgroup is normal?

Detailed: Let $G$ and $H$ be a finite group and its subgroup, respectively. If $H$ is normal in $G$ then the left cosets coincides with the right ones, i.e. $gH=Hg$ for any $g\in G$.

Suppose that for any $a\in G$ there is $b \in G$ such that $aH=Hb$, i.e. left and right cosets 'coincide as partitions of a group into subsets', but it is not required that $aH=Ha$. I wonder if one can derive that $H$ is actually a normal subgroup or give a counter example?

-
If $aH = Hb$, then $a \in Hb$ and hence $Hb = Ha$. – Tom De Medts Sep 15 2011 at 13:44
Indeed, the question was stupid, thanks! – Eugene Starling Sep 15 2011 at 13:50
In my course this morning, I shall teach the definition of a normal subgroup: its left- and right-cosets coincide ... – Denis Serre Sep 16 2011 at 5:30