Permutation IAPS

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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)