If a non-trivial subgroup A of the group of continuous automorphisms of a noncyclic free pro-p group F has finite order, not divisible by p, then the group of fixed points FixF (A) has infinite rank. The semi-direct product F>!A is the universal p-Frattini cover of a finite group G, and so is the projective limit of a sequence of finite groups starting with G, each a canonical group extension o...

In [4], Cameron and Cara showed a relationship between independent generating sets of a group G and RWPri geometries for G. We first notice a connection between such independent generating sets in G and those in the quotient G/Φ(G), where Φ(G) is the Frattini subgroup of G. This suggests a similar connection for RWPri geometries. We prove that there is a one-to-one correspondence between the RW...

The notions of nearly-maximal and near Frattini subgroups considered by J.B. Riles in [20] and the natural related notions are characterized for abelian groups.

We prove that for any prime number p, every finite non-abelian p-group G of class 2 has a noninner automorphism of order p leaving either the Frattini subgroup Φ(G) or Ω1(Z(G)) elementwise fixed.

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups with automorphism group a p-group. Yet the goal of this paper is to prove that almost all finite p-groups do have automorphism group a p-group when p is odd. The asymptotic sense in which the theorem holds involves bounding the Frattini length of the p-gro...