Subgroups generators with respect to group generators of composite order If I have a group $\mathcal{G}$ of order $N=npq$ and subgroups $\mathcal{G_n,G_p,G_q}$ of order $n$, $p$, $g$ respectively and if $g$ is a generator of $\mathcal{G}$ why then $g^{nq}$ is a generator of $\mathcal{G_p}$, $g^{np}$ is a generator of $\mathcal{G_q}$ and $g^{pq}$ is a generator of $\mathcal{G_n}$ ? I am trying to understand a research paper
If $\mathcal{G}$ is of order $N$ (who doesn't look like a prime number btw) and $g$ is a generator of $\mathcal{G}$ then $g$ has order N. Since $(g^{nq})^p=1$ and $\forall 1\le k &lt;p, (g^{nq})^k\neq 1$ (g has order $N$ and $knq&lt;N$) then $g^{nq}$ generates a subgroup of $\mathcal{G}$ of order $p$ and there's only one such subgroup : $\mathcal{G_p}$.
The reasoning is the same for the rest.
Hope it's clear and correct