I am trying to write the presentation of the symmetric group $S_{5}$. We know that $S_{5}$ is generated by $a=(1,2)$ and $b=(1,2,3,4,5)$. Using this I am trying to write presentation of $S_{5}$.
My attempt is $$S_{5}= \langle a,b : a^{2}, b^{5},(ab)^{4}, (ab^{2})^{6},(ab^{3})^{6},(ab^{4})^{4} \rangle.$$
But I am confused. Is this right or not? Please give me suggestions, or write the correct short form.
What you have written is not a presentation. In general it is not a good or efficient strategy to just write down powers of elements hoping that you will get a presentation. The best known presentation of $S_5$ is on the generators $a=(1,2)$, $b=(2,3)$, $c=(3,4)$, $d=(4,5)$: $$\langle a,b,c,d \mid a^2,b^2,c^2,d^2,(ab)^3,(bc)^3,(cd)^3,(ac)^2,(ad)^2,(cd)^2 \rangle.$$ If you really need a presentation on the generators $a=(1,2)$, $b=(1,2,3,4,5)$, then I can easily find one on a computer: $$\langle a,b \mid a^2,b^5,(ab)^4,(bab^{-2}ab)^2 \rangle,$$ which you could rewrite as $$\langle a,b \mid a^2,b^5,(ab)^4,[a,b^2]^2 \rangle.$$