Automorfismos dos 2-grupos de Suzuki

Abstract

Suzuki 2-groups form an interesting class of finite 2-groups. They were introduced by Higman in 1961 and further studied by various authors. By definition, if G is a Suzuki 2-group, then a solvable subgroup of Aut(G) permutes transitively the involutions of G. Higman identified four infinite families of Suzuki 2-groups and proved that each Suzuki 2-group belongs, up to isomorphism, to one of these families. This dissertation is devoted to the study of the automorphisms of Suzuki 2-groups. The main theorems describes the automorphism groups of the groups A(...) and B(n) (the latter is isomorphic to a Sylow 2-subgroup of SU(...). The main result states that in these cases the automorphism groups are isomorphic to the semidirect product of an elementary abelian 2-group and a group isomorphic to (...) where m = n in the case of A(...) and m = 2n in the case of B(n). The description of the automorphism groups is obtained using a methodology based on the theory of permutation groups and linear groups. The novel idea in the proof presented here for the groups A(..) is the use of the characterization by Kantor of the linear groups that contain a Singer cycle. In the case of B(n), we adopt the proof presented by Landrock in 1974, which is also based on the theory of Singer cycles and on a result by Hawkes that describe a certain part of the automorphism group of a 2-group. We obtain, as a by-product, a result that states that the Suzuki 2-groups that we study have precisely 3 characteristic subgroups, and thus we partially verify a conjecture made by Glasby, Palfy and Schneider in 2011.