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The Commuting and Cyclic Graphs of Solvable A-Groups

Carleton, Rachel
http://orcid.org/0000-0002-0708-9963
http://rave.ohiolink.edu/etdc/view?acc_num=kent1713384046910533

Abstract Details

2024, PHD, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
The commuting graph of a group is a graph whose vertices are the noncentral elements of the group, and two vertices are connected in the commuting graph if the elements commute. We first investigate the commuting graph of finite, solvable A-groups, groups whose Sylow subgroups are abelian. We determine when the commuting graph of a solvable A-group will be connected and prove that, when connected, the diameter of the commuting graph will be at most 6. Next, we briefly turn our attention to commuting graphs of p-groups, where p is a prime. We build off work that established there was no universal upper bound on the diameter of the commuting graph by constructing a family of p-groups whose commuting graphs have increasing diameters. Lastly, we define the cyclic graph of a group to be the graph whose vertices are the nontrivial elements of a group, and two vertices are connected in the cyclic graph if the elements generate a cyclic subgroup. We investigate the cyclic graph of a finite, solvable A-group and establish an upper bound for the diameter. More specifically, if Z(Gi), where Gi is the i-th term in the derived series, we establish that when the deleted enhanced power graph is connected, it will have diameter at most 6+4i. For A-groups of derived length 2, we prove an even stronger bound of 8 for the diameter.
Mark Lewis (Advisor)
Stephen Gagola, Jr. (Committee Member)
Hamza Balci (Committee Member)
Joanne Caniglia (Committee Member)
Donald White (Committee Member)
64 p.
Mathematics
commuting graph; solvable groups; finite group theory; A-groups; cyclic graph; deleted enhanced power graph

Recommended Citations

Citations

  • Carleton, R. (2024). The Commuting and Cyclic Graphs of Solvable A-Groups [Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1713384046910533

    APA Style (7th edition)

  • Carleton, Rachel. The Commuting and Cyclic Graphs of Solvable A-Groups. 2024. Kent State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1713384046910533.

    MLA Style (8th edition)

  • Carleton, Rachel. "The Commuting and Cyclic Graphs of Solvable A-Groups." Doctoral dissertation, Kent State University, 2024. http://rave.ohiolink.edu/etdc/view?acc_num=kent1713384046910533

    Chicago Manual of Style (17th edition)

kent1713384046910533
82
© 2024, all rights reserved.
This open access ETD is published by Kent State University and OhioLINK.