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Author
Title Orders of Perfect Groups with Dihedral Involution Centralizers
URL
Publication Date
Degree MS
Discipline/Department Mathematics
Degree Level masters
University/Publisher University of Akron
Abstract Let G be a finite group that is equal to its commutator subgroup, and suppose that G contains an element of order 2 whose centralizer in G is dihedral of 2-power order. We study the cases where this centralizer is dihedral of order 8, 16, 32, 64, 128, or 256. It is true in each case that this centralizer is a Sylow 2-subgroup of G. We then use character-theoretic techniques to generate a list of possibilities for the order of G. In the process of generating this list of possible orders, we prove several results about the structure of our group under consideration. We then strengthen the original hypotheses to require G to be non-abelian simple, and we use the results proved about the structure of G to eliminate all possible orders such that there is no non-abelian simple group of that order.
Subjects/Keywords Mathematics; simple groups; dihedral groups; involutions; projective special linear groups; character theory
Contributors Riedl, Jeffrey (Advisor)
Language en
Rights unrestricted ; This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.
Country of Publication us
Format application/pdf
Record ID oai:etd.ohiolink.edu:akron1365259761
Repository ohiolink
Date Retrieved
Date Indexed 2020-10-19
Grantor University of Akron

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…finite groups. Hence, the study and classification of finite simple groups has been one of the most fundamental problems in the theory of finite groups. For an arbitrary prime power q, the special linear group SL(2, q) consists of all the…

…invertible 2 2 matrices with determinant 1 over the field of q elements. The projective special linear group P SL(2, q) is defined as P SL(2, q) = SL(2, q)/Z(SL(2, q)), 1 where Z(SL(2, q)…

…76 B.10 The case for n = 7, b = 1 or b = 2 . . . . . . . . . . . . . . . . . . . . . 76 viii CHAPTER I INTRODUCTION All groups in this thesis are considered to be finite. Given a nontrivial group G, we say that a composition series for G is a…

…x28;after a possible re-ordering of the composition factors). Since every group is made up of a unique set of simple composition factors (up to isomorphism), it is understood that finite simple groups form the building blocks of all…

…x29; is the center of SL(2, q). We note that P SL(2, q) is a non-abelian simple group for a prime power q ≥ 4. These groups are the underlying groups at the heart of this thesis, as we now explain. In this thesis we generalize…

…8, then |G| = 168 or 360. We note that the non-abelian simple groups P SL(2, 7) and P SL(2, 9) of orders 168 and 360, respectively, satisfy the hypotheses in Theorem 7.10. The proof found in [1] and our proof of the…

…group orders given in Theorem 1.0.2 can be seen in this context. The group orders listed in the statement of this theorem correspond to the simple groups P SL(2, 7), P SL(2, 9), P SL(2, 17), P SL(2, 31), P SL…

…x28;2, 127), and P SL(2, 257), respectively. These six groups make up all of the non-abelian simple groups under consideration in this thesis. The remaining thirty-six integers given in Table 1.1 do not correspond the to the order of a non…

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