[Math] group of order $20$ has a normal subgroup of order $5$

abstract-algebragroup-theory

Let $G$ be a group of order 20, Prove that $G$ has a normal subgroup of order 5.

Obviously by Sylow theorem there is a subgroup of order 5, and since all Sylow p-subgroups are conjugate the only problem is to show that there is only one sungroup of order 5.

any help would be appreciated

Best Answer

$|G|=20=5\cdot 2^2$. Now, let $n_p(G)$ be the number of Sylow $p$-subgroups of $G$.

Then, Sylow III says

  • $n_5(G)\equiv 1\bmod 5$
  • $n_5(G)$ divides $4$

So, $n_5(G)=1$. This means there is only one Sylow $5$-subgroup of $G$; which in turn, is a normal subgroup of $G$.

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