Is there any group with the non-trivial center subgroup and the nontrivial Frattini subgroup such that intersection of them is trivial?


$\Phi(G)$ and $Z(G)$ denotes Frattini and center of $G$.

If $H,K$ are finite groups then

(Miller) $\Phi(H\times K)=\Phi(H)\times \Phi(K)$.

(Obvious, holds for infinite groups also) $Z(H\times K)=Z(H)\times Z(K)$.

To find an example of a group in question, consider $H$ and $K$ with property that $$Z(H)=1, \Phi(H)\neq 1 ,\hskip5mm \mbox{and} \hskip5mm Z(K)\neq 1, \Phi(K)=1.$$

It is easy to find such $K$. Take $K=\mathbb{Z}_p$.

For $H$, consider dihedral group of order $18$; it can be shown that its Frattini subgroup has order $3$ and center is trivial.

For this $H$ and $K$, consider $G=H\times K$.


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