Any answer will be greatly appreciated!
In the proof of Theorem 1 of the paper,
The number of conjugacy classes of non-normal cyclic subgroups in nilpotent groups of odd order, Journal of Group Theory. Volume 1, Issue 2, Pages 165–171,
$G$ is a finite nilpotent group and $\nu^*(G)$ and $\Phi(G)$ denote the number of conjugacy classes of non-normal cyclic subgroups of $G$ and the Frattini subgroup of $G$, respectively. I have the following questions in step (d) in this proof:
- Why do we have $U=\langle u,w\rangle$?
2.Why is $\langle w,\Phi(U)\rangle$ a maximal subgroup of $G$?
3.Why $x\notin\langle w,\Phi(U)\rangle$?