Permutative APS

From Groupprops
Jump to: navigation, search
This term is related to: APS theory
View other terms related to APS theory | View facts related to APS theory


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 \le j \le m


\sigma(j) = j - m, \qquad m + 1 \le j \le 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)