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 ${\mathrm{\Phi }}_{m,n}:S_m\times S_n\to S_{m+n}$ is defined as follows:

Given a permutation $g\in S_m$ and a permutation $h\in S_n$, the permutation ${\mathrm{\Phi }}_{m,n}(g,h)$ 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:

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