In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.
We introduce and study the lattice of normal subgroups of a group G that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of G, see [Kaplan G., Levy D., Solitary subgroups, Comm. Algebra, 2009, 37(6), 1873–1883]. A precise description of this lattice is given for some particular classes of finite groups.
We assign to each pair of positive integers and a digraph whose set of vertices is and for which there is a directed edge from to if . We investigate the structure of . In particular, upper bounds are given for the longest cycle in . We find subdigraphs of , called fundamental constituents of , for which all trees attached to cycle vertices are isomorphic.