Permutation IAPS: Difference between revisions

Latest revision as of 19:15, 18 August 2008

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 $n^{t h}$ 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 $n^{t h}$ member is $S_{n}$ and the block concatenation map $Φ_{m, n} : S_{m} \times S_{n} \to S_{m + n}$ is defined as follows:

Given a permutation $g \in S_{m}$ and a permutation $h \in S_{n}$ , the permutation $Φ_{m, n} (g, h)$ 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:

$Φ_{3, 4} (g, h) = (1, 3) (4, 6, 7)$

@@ Line 5: / Line 5: @@
 ===Symbol-free definition===
-The permutation IAPS is an [[IAPS of groups]] where the <math>n^{th}</math> member is the symmetric group <math>S_n</math>, and where the block concatenation map <math>S_m</math> &times; <math>S_n</math> &rarr; S_{m+n} is defined as the permutation that permutes the first <math>m</math> symbols according to the left argument and the next <math>n</math> symbols according to the second argument.
+The permutation IAPS is an [[IAPS of groups]] where the <math>n^{th}</math> member is the symmetric group <math>S_n</math>, and where the block concatenation map <math>S_m \times S_n \to S_{m+n}</math> is defined as the permutation that permutes the first <math>m</math> symbols according to the permutation in <math>S_m</math> and the next <math>n</math> symbols according to the permutation in <math>S_n</math>.
 ===Definition with symbols===
@@ Line 11: / Line 11: @@
 The permutation IAPS is an [[IAPS of groups]] where the <math>n^{th}</math> member is <math>S_n</math> and the block concatenation map <math>\Phi_{m,n}: S_m \times S_n \to S_{m+n}</math> is defined as follows:
-{{fillin}}
+Given a permutation <math>g \in S_m</math> and a permutation <math>h \in S_n</math>, the permutation <math>\Phi_{m,n}(g,h)</math> is defined as the following permutation on <math>\{1,2,3,\dots,m+n\}</math>. It sends <math>i \in \{ 1,2,3,\dots,m \}</math> to <math>g(i)</math>, and sends <math>j \in \{m + 1, m+2, \dots,m+n\}</math> to <math>h(j-m) + m</math>.
+==Examples==
+As an example, consider <math>m = 3</math>, <math>n = 4</math>. Let <math>g = (1,3)</math>, and <math>h = (1,3,4)</math>. Then:
+<math>\Phi_{3,4}(g,h) = (1,3)(4,6,7)</math>