Prime and N-Prime graphs of solvable uniformly semi-rational groups

Abstract

In this paper, we study the prime graph, also known as Gruenberg-Kegel graph, realizable by finite solvable uniformly semi-rational groups. This is primarily achieved by classifying the prime graphs realizable by metanilpotent groups. We also introduce a new class of groups, called MP* groups, which extends the class of groups with Magnus property and forms a natural subclass of uniformly semi-rational groups. For MP* groups, we give a complete classification of the prime graphs realized by them. In the process, we obtain the realizability of prime graphs by uniformly semi-rational groups, assuming them to be in varied substantial subclasses of solvable groups, namely, 2-Frobenius, metacyclic, metabelian, nilpotent-by-abelian, abelian-by-cyclic and cyclic-by-abelian. We also classify the so called N-prime graphs for all these classes of groups. The prime graph question for uniformly semi-rational groups has also been addressed.