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Summers FINAL 4 30 2021 with cert.pdf (303.4 KB)

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The Influence of Subgroup Structure on Finite Groups Which are the Product of Two Subgroups

Summers, Andrew
http://rave.ohiolink.edu/etdc/view?acc_num=ysu1620058166407179

Abstract Details

2021, Master of Science in Mathematics, Youngstown State University, Department of Mathematics and Statistics.
In group theory, it is often the case that a group can be written as the product of two of its subgroups. Take for example S3, the symmetric group on a set of three elements, which can be written as S3 = A3 〈 (12) 〉 or alternatively D4, the group of symmetries of a square, which can be written as D4 = 〈 (1234) 〉 〈 (13) 〉. It is therefore natural to wonder what influence the structures of these subgroups have on the structure of the group as a whole. For example, if G is a group, H ≤ G and K ≤ G such that G = HK, where both H and K are cyclic, one may ask if G is consequently cyclic as well. Moreover, if G is not cyclic, then what, if anything, can be said about its structure? In actuality, it happens that G is, in fact, not cyclic, but solvable. In this master's thesis we establish several important classes of groups which will be used to explore the influence of subgroup structure on groups which are the product of two subgroups. Additionally, we will lead up to the strongest possible statement about the structure of such groups, without placing additional constraints on H and K. This result was originally proved by Helmut Wielandt in 1958 under the assumption that the orders of H and K were coprime. These assumptions were later dropped in an improved result by Otto Kegel in 1961.
Neil Flowers, PhD (Advisor)
Thomas Madsen, PhD (Committee Member)
Thomas Wakefield, PhD (Committee Member)
65 p.
Mathematics
Mathematics; Group Theory; Product of Two Subgroups; Finite Groups

Recommended Citations

Citations

  • Summers, A. (2021). The Influence of Subgroup Structure on Finite Groups Which are the Product of Two Subgroups [Master's thesis, Youngstown State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1620058166407179

    APA Style (7th edition)

  • Summers, Andrew. The Influence of Subgroup Structure on Finite Groups Which are the Product of Two Subgroups. 2021. Youngstown State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ysu1620058166407179.

    MLA Style (8th edition)

  • Summers, Andrew. "The Influence of Subgroup Structure on Finite Groups Which are the Product of Two Subgroups." Master's thesis, Youngstown State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1620058166407179

    Chicago Manual of Style (17th edition)

ysu1620058166407179
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© 2021, all rights reserved.
This open access ETD is published by Youngstown State University and OhioLINK.