A nonembedding theorem for finite groups

Abstract:

Let $N$ be the class of nilpotent groups with the following properties: (1) The center of $N,{Z_ \bot }(N)$ is of prime order. (2) There exists an abelian characteristic subgroup $A$ of $N$ such that ${Z_1}(N) \subset A \subseteq {Z_2}(N)$ where ${Z_2}(N)$ is the second term in the upper central series of $N$. The main result shown is the following: $N \in \mathfrak {X}$, then $N$ cannot be an invariant subgroup contained in the Frattini subgroup of a finite group.