Let $N_{ g,n}$ be a genus $g$ compact non-orientable surface with $n$ boundaries. We explain about relations on the level $d$ mapping class group $\mathcal{ M}_d(N_{ g,0})$ of $N_{ g,0}$ and the level $d$ principal congruence subgroup $\Gamma_d(g-1)$ of $\mathrm{ SL}(g-1;\mathbb{ Z})$. As applications, we give a normal generating set of $\mathcal{ M}_d(N_{ g,n})$ for $g\ge4$ and $n\ge0$, and finite generating sets of $\mathcal{ M}_d(N_{ g,n})$ for some $d$, any $g\ge4$ and $n\ge0$.
Categories: math.GT