Group Theory Seminar

The Gruenberg-Kegel graph of rational and inverse semirational groups

Speaker:  Ángel del Río (Universidad de Murcia)
Date:  Friday, 02 June 2023 - 11:30
Place:  Aula Naranja, ICMAT


The Gruenberg-Kegel graph (abbreviated GK-graph) of a group G is the loop-free undirected graph whose vertices are the prime integers which occurs as the order of an element of G, and two different vertices p and q are joined by an edge if G has an element of order pq. The GK-graph of G provides relevant information of G.

A fundamental group theoretical problem consists in determining which graphs occur as GK-graphs of groups in a certain family. For example every graph is the GK-graph of some group but the GK-graphs of finite groups have at most 6 connected components. The GK-graphs of finite solvable groups have recently been classified.

Recall that a finite group is said to be rational if the character table is formed by rational numbers, or equivalently every two elements generating the same subgroup are conjugate.
A weaker condition is that of inverse semirational groups which are groups in which two elements g and h generating the same subgroup are conjugate or g and h^{-1} are. In the terminology of "group ringers'' inverse semirational groups are called cut because they are precisely the finite groups such that every Central Unit of its integral group ring is Trivial i.e. belong to G or -G.

In this talk we will present some recent result on the GK-graphs of rational and inverse semirational groups.