# Sub-APS of groups

Let $(G,\Phi)$ be an APS of groups. A sub-APS $H$ of $G$ is, for every $n$, a subgroup $H_n$ of $G_n$ such that $\Phi_{m,n}(g,h) \in H_{m+n}$ whenever $g \in H_m, h \in H_n$. Thus, $H$ can be viewed as an APS of groups, in its own right.