https://doi.org/10.31219/osf.io/r3jvu

Published 15 Apr 2021 Volume 3, Article 227 (IX)

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1. Abstract

We calculate all compositions of permutations in the symmetric groups $S_2$ and $S_3$.

2. The symmetric group

$n\in\mathbb N=\{1,2,3,...\}$

$\mathbb N_n=\{1,2,3,...,n-1,n\}$

$S_n:=$ group of all the permutations of $\mathbb N_n$ (symmetric group)

$S_1=\{(1)\}$

$S_2=\{(1),(12)\}$

$S_3=\{(1),(12),(13),(23),(123),(132)\}$

$\vdots$

3. Composition of permutations

Notation

$xfg:=g(f(x))$

S2

ff, fg

S3