Periodic structures in the graph associated with p-groups of maximal class [Elektronische Ressource] / von Heiko Dietrich

Periodic structures in the graph associated with p-groups of maximalclassVon der Carl-Friedrich-Gauß-Fakult¨at derTechnischen Universit¨at Carolo-Wilhelmina zu Braunschweigzur Erlangung des GradesDoktor der Naturwissenschaften (Dr. rer. nat.)genehmigte Dissertation vonHeiko Dietrichgeboren am 15. April 1980in Langen (Hessen)Eingereicht am: 27.04.2009Mu¨ndliche Pru¨fung am: 26.06.2009Referentin: Prof. Dr. Bettina EickKorreferent: Prof. Dr. Gerhard Hiß2009iiDedicated to my grandfatherHans Viktor KleinivSummaryA finite group whose order is a power of a prime p is called a finite p-group. Among finitegroups, p-groups take a special position: For example, every finite group contains large p-groups as subgroups by the Sylow Theorems. The classification of p-groups is a difficultproblem and, in general, not even the exact number of isomorphism types of groups of ordernp is known. The asymptotic estimates of Higman (1960) and Sims (1965) show that there3 8/32n /27+O(n ) nare p isomorphism types of groups of order p . Higman’s PORC Conjecturen(1960) claims that for fixed n the number of isomorphism types of groups of order p is apolynomial on residue classes.A special type ofp-groups arep-groups of maximal class: These are thep-groups of ordernp with nilpotency class n−1. A first major study of maximal class groups was carriedout by Blackburn in 1958.