Permutative APS

This term is related to: APS theory
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Definition

A permutative APS of groups is an APS of groups $(G,\Phi )$ equipped with an APS-on-APS action on it by the permutation IAPS, having the following property:

If $\sigma :\{1,2,3,\dots ,m+n\}\to \{1,2,3,\dots ,m+n\}$ is a permutation defined as:

$\sigma (j)=n+j,\qquad 1\leq j\leq m$

and:

$\sigma (j)=j-m,\qquad m+1\leq j\leq n$

Then, for $g\in G_{m},h\in G_{n}$, we have:

$\Phi _{n,m}(h,g)=\sigma \cdot \Phi _{m,n}(g,h)$