I have a question which I assume is under Group or set theory. I have attempted to answer it but I have now a colleague is saying I am wrong. Need your wisdom on it please.
Question: Consider the element $\alpha = (125)(476)$ and $\beta = (142)(3567)$ of $S_7$. Find $\alpha\beta$, $\beta\alpha$, $\alpha^2$ and $\alpha^{-1}$ (where for x $\in$ {1,2,3,4,5,6,7}, $(\alpha\beta) (x)$ is defined to be $\alpha(\beta(x)))$.
My Answer:
$\alpha\beta$ = {(125,142),(125,3567),(476,142),(476,3567)}
$\beta\alpha$ = {(142,125),(142,476),(3567,125),(3567,476)}
$\alpha^2$ = {(125,125),(125,476),(476,125),(476,476)}
$\alpha^{-1}$ = (476)(125)
I thank you very much for your time and greatly appreciate your help.
If $\alpha = (125)(476)$, then $\alpha$ maps $1\mapsto2$, $2\mapsto5$, $5\mapsto1, 4\mapsto7, 7\mapsto6, 6\mapsto4$, and $3\mapsto3.$
Therefore $\alpha^{-1}$ is given by $(152)(467)$.
$\alpha\beta=(125)(476)(142)(3567)$ maps $1\mapsto4\mapsto7, 2\mapsto1\mapsto2, 3\mapsto5\mapsto1, 4\mapsto2\mapsto5$, $5\mapsto6\mapsto4$,
$6\mapsto7\mapsto6,$ and $7\mapsto3\mapsto3$. Therefore. $\alpha\beta$ is given by $(173)(45)$.
I hope that with this knowledge you can now compute $\beta\alpha$ and $\alpha^2$.
You could also note that $\alpha^3$ is the identity so $\alpha^2=\alpha^{-1}$.