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November 22nd, 2017, 01:01 AM   #1
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Question 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
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November 22nd, 2017, 05:36 AM   #2
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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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