# 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: $\Phi_{3,4}(g,h) = (1,3)(4,6,7)$