Gruenberg-Kegel graphs and their applications.

The prime graph of a group G, also known as the Gruenberg-Kegel graph GK(H), after Karl Gruenberg (1928-2007) and Otto Kegel (1934-), has as its vertex set all the prime divisors of the order |G| and two distinct vertices p, q are adjacent exactly when the group G contains an element of order pq.

A survey of the main properties of Gruenberg-Kegel graphs will be given and some their applications to the finite group theory will be presented.