My Math Forum Smallest subgroup problem

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 November 22nd, 2017, 01:01 AM #1 Member   Joined: Jan 2016 From: / Posts: 43 Thanks: 1 Smallest subgroup problem Hi! While learning group theory I stumbled across a problem I am really interested in learning how to solve, unfortunately I don't have the solution. The problem goes like this : Find the smallest subgroup of symmetric group S4 that contains the following elements : $\displaystyle \bigl(\begin{smallmatrix} 1 & 2 & 3 & 4 \\ 3 & 1 & 4 & 2 \end{smallmatrix}\bigr)$ I've been browsing the web to find the basic concepts needed to solve the problem. According to some post I must find the greatest common divisor of these numbers would that be gcd(1,2,3,4) = 1? Thanks in advance! SuperNova
 November 22nd, 2017, 05:36 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 I don't see any reason for worrying about the "greatest common divisor". Start by looking at powers of the given element: $\begin{pmatrix}1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2\end{pmatrix}^2= \begin{pmatrix}1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{pmatrix}$ That's the identity permutation immediately! So the smallest subgroup of only two members, the identity permutation and $\begin{pmatrix}1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2\end{pmatrix}$ itself.

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