A nonembedding theorem for finite groups

Abstract

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