Author
Malyushitsky, Sergey Zenonovich

Year
2004

Advisor
Harada, Koichiro

Abstract
A 2-group (a group of order a power of 2) is called realizable if it occurs as a Sylow 2-subgroup of a finite simple group. The purpose of this thesis is to study all realizable groups of order up to 2¹⁰. From the classification of all simple groups of finite order we know all realizable groups of order up to 2¹⁰ as Sylow 2-subgroups of known finite simple groups. However without the use of classification determining all realizable 2-groups is very difficult.

In the first part of the thesis we present an argument that produces all realizable groups of order up to 32, by eliminating one by one all 2-groups that can not occur as a Sylow 2-subgroup of a finite simple group. When the number of 2-groups of given order becomes too large to handle it by hand we attempt to use a computer for repetitive checks on a large number of 2-groups.

The second part of the thesis is devoted to describing all realizable 2-groups of order up to 2¹⁰ using the classification of all finite simple groups. We determine the identification number of each group in the Small Groups Library in GAP4 and compute the power-commutator presentation of each realizable group S of type G. We also determine the conjugacy classes of involutions of S and their fusion in G, maximal subgroups of S, maximal abelian and maximal elementary abelian subgroups of S, Z(S), S/Z(S), maximal quotient groups of S and maximal normal extra-special subgroups of S. of parameter estimation are obtained through the maximum likelihood method.

Thesis
Malyushitsky.Sergey.Zenonovich.pdf