The Gruenberg-Kegel graph of some solvable groups.

The Gruenberg-Kegel graph of a group is defined as the graph whose vertices are the primes that appear as orders of elements of the group, and there is an edge between two primes p and q if and only if pq is the order of an element of the group. For solvable groups with small fields of characters, like rational groups or inverse semirstional groups, the set of vertices of this graph is known to be bounded (e.g., for rational solvable groups, only the primes 2,3 and 5 can appear). The classification of the Gruenberg-Kegel graphs of solvable rational and inverse semirational groups was started by Bächle, Kiefer, Maheshwary and del Río, but a few graphs were left unresolved. We managed to complete this classification for solvable rational groups, and made some advances toward the classification for solvable inverse semirational groups. 

This talk is based in a joint work with Sara Cebellán Debón and Ángel Del Rio.