In previous papers [4; 5; 6] we constructed and studied many classes of operations on groups which have the following properties: (1) Associative; (2) Commutative; (3) The resultant group contains isomorphic images of the original factors and is generated by them; (4) In the resultant group the intersection of a factor with normal subgroup generated by the remaining factors is the unit element....

For a finite group G and an inverse-closed generating set C of G, let Aut(G;C) consist those automorphisms which leave invariant. We define Aut(G;C)-invariant normal subgroup Φ(G;C) has the property that, for any generators if we remove from it all elements Φ(G;C), then remaining is still G. The contains Frattini Φ(G) but inclusion may be proper. Cayley graph Cay(G,C) edge-transitive acts trans...

All groups considered in this paper are tacitly assumed to be finite. Recall that a formation 5 is a class of groups which is closed under taking homomorphic images and subdirect products. It is known that, if 5 is a formation, each group G has a smallest normal subgroup whose factor group is in 5Such subgroup is called the ^-residual of G and it is denoted by G. A formation # is said to be sat...

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...