We discuss a group-theoretical generalization of the well-known Gauss formula involving the function that counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.
In this paper we consider finite loops of specific order and we show that certain abelian groups are not isomorphic to inner mapping groups of these loops. By using our results we are able to construct a finite solvable group of order 120 which is not isomorphic to the multiplication group of a finite loop.
The aim of this paper is to introduce the notion of BG-injectors of finite groups and invoke this notion to determine the B-injectors of Sₙ and Aₙ and to prove that they are conjugate. This paper provides a new, more straightforward and constructive proof of a result of Bialostocki which determines the B-injectors of the symmetric and alternating groups.
In this paper we consider finite loops whose inner mapping groups are nilpotent. We first consider the case where the inner mapping group of a loop is the direct product of a dihedral group of order and an abelian group. Our second result deals with the case where is a -loop and is a nilpotent group whose nonabelian Sylow subgroups satisfy a special condition. In both cases it turns out that is centrally nilpotent.