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dissertation.pdf (310.42 KB)
ETD Abstract Container
Abstract Header
Prime Character Degree Graphs of Solvable Groups having Diameter Three
Author Info
Sass, Catherine Bray
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=kent1398110266
Abstract Details
Year and Degree
2014, PHD, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
Abstract
Let G be a finite solvable group and cd(G) the set of character degrees of G. The prime character degree graph Δ(G) is the graph whose vertices are the primes dividing the degrees in cd(G) and there is an edge between two distinct primes p and q if their product pq divides some degree in cd(G). When Δ(G) has diameter three we can partition the vertices ρ(G) into four non-empty disjoint subsets ρ
1
∪ρ
2
∪ρ
3
∪ρ
4
where no prime in ρ
1
is adjacent to any prime in ρ
3
∪ρ
4
; no prime in ρ
4
is adjacent to any prime in ρ
1
∪ρ
2
; every prime in ρ
2
is adjacent to some prime in ρ
3
; every prime in ρ
3
is adjacent to some prime in ρ
2
; and |ρ
1
∪ρ
2
|≤|ρ
3
∪ρ
4
|. We will show that the subset ρ
3
must contain at least three vertices. The solvable group G has exactly one normal nonabelian Sylow p-subgroup P, and the prime p is contained in the subset ρ
3
. The factor group G/P' cannot have a normal nonabelian Sylow subgroup where P' is the derived subgroup of P and so Δ(G/P') must be disconnected. If the subset ρ
1
∪ρ
2
has n vertices, then the subset ρ
3
∪ρ
4
must have 2
n
vertices. The group G must have Fitting height three.
Committee
Mark Lewis (Committee Chair)
Stephen Gagola, Jr (Committee Member)
Donald White (Committee Member)
Joanne Caniglia (Committee Member)
Austin Melton (Committee Member)
Pages
70 p.
Subject Headings
Mathematics
Keywords
character theory, solvable groups, character degree graphs,
Recommended Citations
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Citations
Sass, C. B. (2014).
Prime Character Degree Graphs of Solvable Groups having Diameter Three
[Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1398110266
APA Style (7th edition)
Sass, Catherine.
Prime Character Degree Graphs of Solvable Groups having Diameter Three.
2014. Kent State University, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=kent1398110266.
MLA Style (8th edition)
Sass, Catherine. "Prime Character Degree Graphs of Solvable Groups having Diameter Three." Doctoral dissertation, Kent State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=kent1398110266
Chicago Manual of Style (17th edition)
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Document number:
kent1398110266
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Copyright Info
© 2014, some rights reserved.
Prime Character Degree Graphs of Solvable Groups having Diameter Three by Catherine Bray Sass is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by Kent State University and OhioLINK.