Beauville structures in p-groups

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2016

Öğrenci: ŞÜKRAN GÜL

Danışman: GÜLİN ERCAN

Özet:

Given a finite group G and two elements x, y in G, we denote by Sigma(x,y) the union of all conjugates of the cyclic subgroups generated by x, y and xy. Then G is called a Beauville group of unmixed type if the following conditions hold: (i) G is a 2-generator group. (ii) G has two generating sets {x1,y1} and {x2, y2} such that Sigma (x1, y1) intersection Sigma(x2, y2) is 1. In this case, {x1, y1} and {x2, y2} are said to form a Beauville structure for G. The main purpose of this thesis is to extend the knowledge about Beauville p-groups. We will first discuss the conditions under which a 2-generator p-group with a “nice power structure” is a Beauville group. These conditions are similar to the conditions for an abelian p-group to be a Beauville group. In particular, this result applies to all known families of p-groups with a good behavior with respect to powers: regular p-groups, powerful p-groups and more generally potent p-groups, and (generalized) p-central p-groups. Secondly, we investigate Beauville structures in metabelian thin p-groups and in p-groups of maximal class which are either metabelian, or have a maximal subgroup of class at most 2. We next determine which quotients of the Nottingham group over Fp for an odd prime p are Beauville groups. As a result, we get the first known infinite family of 3-groups admitting a Beauville structure. Finally, we prove a conjecture of Boston: he conjectured that if p is greater than or equal to 5, all p-central quotients of the free group on two generators and of the free product of two cyclic groups of order p are Beauville groups.