Let $G:=A_5$. I want to know why all of the elements of $2$-Sylow subgroups and $3$-Sylow subgroups of $G$ are not commutative?
2026-03-25 03:01:03.1774407663
On the subgroups properties of $A_5$
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Apparently you mean it is not true that (all of the elements of) the Sylow $2$ and $3$ subgroups commute.
For this it would be sufficient to find any pair $x,y$ from the respective subgroups that don't commute...
You had done this... Since $(1 2)(3 4)$ and $(1 2 3)$ don't commute, you're done.