Let $D_{2n}=\langle a,b;a^n,b^2,(ba)^2\rangle$ denote the Dihedral Group of order $2n$. Let $H=\langle a^m,b\rangle$ be a subgroup of $D_{2n}$ of even index $m$ such that $m$ divides $n$. Choose a right transversal (set of representatives of right cosets of $H$ in $G$) $S=\{b^ia^{2j+i}:i=0,1~\text{and}~0\leq j\leq \frac{m}{2}-1\}$. Clearly $S$ is a right loop (right quasigroup with two sided identity) with respect to the operation defined by $\{x\circ y\}=S\cap Hxy$. (see more). I wish to show that $S$ is a group; moreover, $S\cong D_{2.\frac{m}{2}}$.
Is it sufficient to show that $(a^2)^{\frac{m}{2}}=1, (ba)^2=1$ and $(ba\circ a^2)^2=1$ with respect to the operation $\circ$ ?