Let G be a finite group and N$\lhd$G such as p||N| (p dividing the order of N). Then for all P p-sylow subgroup of G, N$\cap$P$\ne{e}$

Question

Decide whether the following staement is true of false If true, prove it. If false, provide a counterexample

Let G be a finite group and N$\lhd$G such as p||N| (p dividing the order of N). Then for all P p-sylow subgroup of G, N$\cap$P$\ne{e}$

I think it's false but I don't know how to prove it.. please help :)

Answer 1

Let $p, q$ be distinct prime numbers. Consider $\mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/q\mathbb{Z}$, $N=\mathbb{Z}/p\mathbb{Z}\times\{0\}$ and $P=0\times\mathbb{Z}/q\mathbb{Z}$.