Vol. 18, No. 1, 1966

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Frattini subgroups and Φ-central groups

Homer Franklin Bechtell, Jr.

Vol. 18 (1966), No. 1, 15–23

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 is a group of automorphisms of a group N, N has an -central series N = N0 > N1 > > Nr = 1 if x1xa Nj for al x Nj1, 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 . Then Φ() = (Φ(G)), (Φ(G)) the inner automorphisrn gromp of Φ(G). Furthermore if G is nilpotent, then each subgroup N Φ(G), N invariant under , possess an -central series. A class of nilpotent groups N is defined as Φ-central provided that N possesses at least one nilpotent group of automorphisms 1 such that Φ() = (N) and N possesses an -central series. Several theorems develop results about Φ-central groups and the associated -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 , a nonabelian subgroup invariant under 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.

Mathematical Subject Classification
Primary: 20.25
Received: 13 February 1965
Published: 1 July 1966
Homer Franklin Bechtell, Jr.