Find $\alpha,\beta,\gamma\in S_4$ such that $\alpha\beta=\beta\alpha$, $\beta\gamma=\gamma\beta$ but $\alpha\gamma\neq\gamma\alpha$.

Question

Can you give an example for $\alpha,\beta,\gamma\in S_4$ (permutation group for the set $\{1,2,3,4\}$) such that $\alpha\beta=\beta\alpha$, $\beta\gamma=\gamma\beta$ but $\alpha\gamma\neq\gamma\alpha$. I have tried many times but failed.

What have you tried? What is $S_4$ group, there is no standard notation for stuff like this and you can't assume everyone uses the same convention. — Bertrand Wittgenstein's Ghost, Dec 21 '22 at 05:46
@BertrandWittgenstein'sGhost: to be fair, $S_4$ is standard notation (at least in this context). — hello, Dec 21 '22 at 06:04
@hello Perhaps it is. Perhaps it's not. That's besides the point. The comment was a general hint at how Math SE questions should be formatted: Work needs to be shown, and it's better to provide all the relevant details. — Bertrand Wittgenstein's Ghost, Dec 21 '22 at 08:19

score -1 · Accepted Answer · answered Dec 21 '22 at 06:11

-1

$\alpha=(12),\beta =(34)(12),\gamma=(13)(24).$

$\alpha \beta \alpha =\beta .$

$\beta\gamma \beta=\gamma .$

$\alpha \gamma \alpha =(23)(14).$

answered Dec 21 '22 at 06:11

calc ll

8,427

1

Please don't encourage low quality questions. – Bertrand Wittgenstein's Ghost Dec 21 '22 at 08:20

Find $\alpha,\beta,\gamma\in S_4$ such that $\alpha\beta=\beta\alpha$, $\beta\gamma=\gamma\beta$ but $\alpha\gamma\neq\gamma\alpha$.

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