Graduate How many cosets are there when taking a subgroup in a group and forming cosets?

[LagrangeEuler]

When we take some subgroup $H$ in $G$. And form cosets $g_1H, g_2H,...,g_{n}H$. Is $H$ also coset $eH$, where $e$ is neutral? So do we have here $n$ or $n+1$ cosets?

 

[fresh_42]

When we take some subgroup $H$ in $G$. And form cosets $g_1H, g_2H,...,g_{n}H$. Is $H$ also coset $eH$, where $e$ is neutral? So do we have here $n$ or $n+1$ cosets?

That depends on how you count?
If $G = g_1H \cup \ldots \cup g_nH$ then we have $n$ cosets. Now $eH=g_iH$ for some $i$, so we can choose $g_i=e$ and renumber them as e.g. $g_1, \ldots , g_{n-1},e$.
If $G=g_1H \cup \ldots \cup g_nH\cup eH$ then we have $n+1$ cosets.
The total number of cosets in a finite group is $|G/H|=|G|:|H|$. Now define $n$. It is usually more convenient to have $n\,\vert \,|G|$ than to have $(n+1) \,\vert \,|G|$, but finally it's up to you.