4.4 Let $G$ be a finite p-group; show that if $H$ is a normal subgroup of G having order $p$, then $H$ is a subgroup of $Z(G)$.
$Z(G)=\left\{x\in G| xy=yx, \forall y\in G \right\}.$
Can you help me solve it and explain in detail? Thank you very much. Good health!
Use the previous exercise (4.3), which states that $K\cap Z(G)\ne 1$ whenever $K\lhd G$. Now $H\cap Z(G)\ne 1$, and so $H\cap Z(G)$ has order $p$. So $H\cap Z(G) = H$.
Hint of Exercise 4.3: $K\lhd G$ implies that $K$ is a union of some conjugacy classes of $G$.