Pointwise nonlinearities, neighborhood pooling, subgroup sampling, and coset pooling

If $gH=g^{\mathrm{\prime }}H$, then $g^{\mathrm{\prime }}=g^{\mathrm{\prime }}e\in g^{\mathrm{\prime }}H=gH$, so [F2] gives $g^{\mathrm{\prime }}=g{h}_0$ for some $h_0\in H$. By [F1], $h_{0}H\subseteq H$; conversely, every $k\in H$ equals $h_0(h_0^{-1}k)$ with $h_0^{-1}k\in H$, so $H\subseteq h_{0}H$. Hence $h_{0}H=H$ and $(a{g}^{\mathrm{\prime }})H=(ag{h}_0)H=ag(h_{0}H)=agH$, proving that $a\cdot (gH)=(ag)H$ is independent of the representative. It satisfies the identity and composition laws in [F3] because $eg=g$ and $(ab)g=a(bg)$. Thus it is a left action on $G\mathrm{/}H$.

If $g^{\mathrm{\prime }}=g{h}_0$ with $h_0\in H$, then $h_{0}H=H$: closure gives $h_{0}H\subseteq H$, and for $k\in H$, $k=h_0(h_0^{-1}k)\in h_{0}H$ by [F1]. Hence $\{g^{\mathrm{\prime }}h:h\in H\}=\{gh:h\in H\}$, so the componentwise maximum defining $Q_{H}F(gH)$ is independent of the representative.

By [F1], $H$ contains the identity, so $H$ and the given set $U$ are nonempty and finite. Each displayed maximum therefore exists in $\mathbb{R}$; the pointwise operation and subgroup restriction are also functions on their stated domains. Thus all four operators in the Definition are well-defined.