Permutation IAPS

This article describes a particular IAPS of groups, or family of such IAPSes parametrized by some structure

Definition

Symbol-free definition

The permutation IAPS is an IAPS of groups where the $n^{th}$ member is the symmetric group $S_{n}$, and where the block concatenation map $S_{m}\times S_{n}\to S_{m+n}$ is defined as the permutation that permutes the first $m$ symbols according to the permutation in $S_{m}$ and the next $n$ symbols according to the permutation in $S_{n}$.

Definition with symbols

The permutation IAPS is an IAPS of groups where the $n^{th}$ member is $S_{n}$ and the block concatenation map $\Phi _{m,n}:S_{m}\times S_{n}\to S_{m+n}$ is defined as follows:

Given a permutation $g\in S_{m}$ and a permutation $h\in S_{n}$, the permutation $\Phi _{m,n}(g,h)$ is defined as the following permutation on $\{1,2,3,\dots ,m+n\}$. It sends $i\in \{1,2,3,\dots ,m\}$ to $g(i)$, and sends $j\in \{m+1,m+2,\dots ,m+n\}$ to $h(j-m)+m$.

Examples

As an example, consider $m=3$, $n=4$. Let $g=(1,3)$, and $h=(1,3,4)$. Then:

$\Phi _{3,4}(g,h)=(1,3)(4,6,7)$