Let $\displaystyle |G|=p^{\alpha}m, \ p\nmid m$. Then $\displaystyle Syl_p(G)\neq\O$ that is $\displaystyle n_p\geq 1$ where $\displaystyle n_p$ number of Sylow p subgroups.

Let G be a minimal counterexample.

$\displaystyle p\mid |Z(G)| \ \ \text{or} \ \ p\nmid |Z(G)|$

Why are me looking at p dividing or not dividing the center of G?