Raziskovalni matematični seminar - Arhiv
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.
A finite group H is called a P-group for some set of prime numbers P when all primes that divide the order |H| lie in P. The P-subgroup H in G is its Hall subgroup iff besides this the index |G:H| is not divided by any prime p from the set P. We will focus on the structure of normal series in finite groups with Hall maximal subgroups and in finite groups which are prime spectrum minimal. Some additional properties of such groups will also be discussed.