Φ-central groups are
introduced as a step in the direction of determining sufficiency conditions for a group
to be the Frattini subgroup of some finite p-group and the related extension problem.
The notion of Φ-centrality arises by uniting the concept of an E-group with the
generalized central series of Kaloujnine. An E-group is defined as a finite group G
such that Φ(N) ≦ Φ(G) for each subgroup N ≦ G. If ℋ is a group of automorphisms
of a group N, N has an ℋ-central series N = N0> N1>⋯> Nr= 1 if x−1xa∈ Nj
for al x ∈ Nj−1, all a ∈ℋ , xa the image of x under the automorphism a ∈ℋ,
j = 0,1,⋯,r − 1.
Denote the automorphism group induced on Φ(G) by transformation of elements
of an E-group G by ℋ. Then Φ(ℋ) = ℐ(Φ(G)), ℐ(Φ(G)) the inner automorphisrn
gromp of Φ(G). Furthermore if G is nilpotent, then each subgroup N ≦ Φ(G), N
invariant under ℋ, possess an ℋ-central series. A class of nilpotent groups N is
defined as Φ-central provided that N possesses at least one nilpotent group of
automorphisms ℋ≠1 such that Φ(ℋ) = ℐ(N) and N possesses an ℋ-central series.
Several theorems develop results about Φ-central groups and the associated ℋ-central
series analogous to those between nilpotent groups and their associated central series.
Then it is shown that in a p-group, Φ-central with respect to a p-group of
automorphism ℋ, a nonabelian subgroup invariant under ℋ cannot have a
cyclic center. The paper concludes with the permissible types of nonabelian
groups of order p4 that can be Φ-central with respect to a nontrivial group of
p-automorphiszns.
