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# Classification of finite simple groups of small order

Patricija Podboj (2022) Classification of finite simple groups of small order. MSc thesis.  Preview
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## Abstract

At the end of the 19th century Otto Hölder proposed one of the most important problems in group theory, the mathematical field this master’s thesis belongs to. Part of the problem represents the classification of finite simple groups. Finite simple groups are finite nontrivial groups whose only normal subgroups are the trivial group and the group itself. There were many mathematicians that participated in solving Hölder’s problem which was finally resolved almost a century later. This master’s thesis focuses on a small part of the mentioned classification. We want to find all finite simple groups of order no more than 200. All groups of prime order are simple. It turns out that those are also the only commutative simple groups, which follows directly from Lagrange and Cauchy theorems. Therefore, we want to find all noncommutative finite simple groups of order no more than 200. The most important step in achieving this goal in this master’s thesis are two propositions that help us eliminate orders of special forms. Together with all primes there are 189 orders of such form. We prove those propositions using some important structural results from group theory (e.g. Sylow theorems, Cauchy and Lagrange theorems, group actions) and we then determine the corresponding orders using the MAGMA software. To find all finite simple groups, mathematicians have been using many more complex results. That is why this master’s thesis also demonstrates the usefulness of the two propositions we used for eliminating higher orders. After eliminating most of the 200 orders by using the mentioned propositions, there are only 11 orders left. We show that there are no finite simple groups for 9 of those orders by using the above results and some additional arguments. It turns out that there are exactly two noncommutative finite simple groups of order no more than 200. These groups are A5 of order 60 and PSL(2, 7) of order 168. They are part of two very important infinite families of groups, the alternating groups An and the projective special linear groups PSL(n, q), most of which are simple groups. In the master’s thesis we prove the simplicity of the mentioned two groups. We also prove that the group A5 is the only simple group of order 60 and is therefore the smallest noncommutative finite simple group.

Item Type: Thesis (MSc thesis)
Keywords: simple group, normal subgroup, order of the group, classification of finite simple groups, Sylow theorems
Number of Pages: 66
Language of Content: Slovenian
Mentor / Comentors:
Mentor / ComentorsIDFunction
izr. prof. dr. Primož ŠparlMentor
Link to COBISS: https://plus.si.cobiss.net/opac7/bib/peflj/105458691
Institution: University of Ljubljana
Department: Faculty of Education
Item ID: 7163
Date Deposited: 21 Apr 2022 08:26