Full Record

Author | Strayer, Michael Christopher |

Title | Orders of Perfect Groups with Dihedral Involution Centralizers |

URL | http://rave.ohiolink.edu/etdc/view?acc_num=akron1365259761 |

Publication Date | 2013 |

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 | 2020-10-17 |

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 â‰¥ 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…