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Classification of the Prime Graphs of $\operatorname{Sz}(8)$-, $\operatorname{Sz}(32)$-, and $\operatorname{PSL}(2, 2^5)$-Solvable Groups

Thomas Michael Keller, Zachary Martin, Alexa Renner, Gabriel Roca, Eric Yu

Abstract

For a finite group $G$, the vertices of the prime graph $Γ(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if there is an element of order $pq$ in $G$. Prime graphs of solvable groups have been classified, and prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group $T$ have been classified in the case where $T$ has order divisible by exactly three or four distinct primes, except for the cases $T = \operatorname{Sz}(8)$, $T = \operatorname{Sz}(32)$, and $T = \operatorname{PSL}(2,q)$, which in some sense are the hardest cases. In this paper, we complete the classification for $T = \operatorname{Sz}(32)$, $T = \operatorname{Sz}(8)$, and $T = \operatorname{PSL}(2,2^5)$, with the latter two being the first cases ever studied where $|\text{Out}(T)|$ has prime factors which do not divide $|T|$. The groups studied in this paper are also the first ones requiring knowledge of their Brauer character tables to complete the classification task.

Classification of the Prime Graphs of $\operatorname{Sz}(8)$-, $\operatorname{Sz}(32)$-, and $\operatorname{PSL}(2, 2^5)$-Solvable Groups

Abstract

For a finite group , the vertices of the prime graph are the primes that divide , and two vertices and are connected by an edge if there is an element of order in . Prime graphs of solvable groups have been classified, and prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group have been classified in the case where has order divisible by exactly three or four distinct primes, except for the cases , , and , which in some sense are the hardest cases. In this paper, we complete the classification for , , and , with the latter two being the first cases ever studied where has prime factors which do not divide . The groups studied in this paper are also the first ones requiring knowledge of their Brauer character tables to complete the classification task.

Paper Structure

This paper contains 9 sections, 70 theorems, 13 equations, 9 figures.

Table of Contents

  1. Prime Graphs of Abelian Extensions
  2. The Suzuki Group Sz(32)
  3. Reduction to Irreducible Representations
  4. The Subgraphs Induced by $\pi(T)$ and $\pi(G) \setminus \pi(T)$
  5. General Lemmas
  6. The Suzuki Group Sz(8)
  7. The Projective Special Linear Group PSL(2,$\mathbf{2^5}$)
  8. Outlook
  9. Acknowledgements

Key Result

Theorem 1

2015REU An unlabeled simple graph $\Xi$ is isomorphic to the prime graph complement of a solvable group if and only if it is 3-colorable and triangle-free.

Figures (9)

  • Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits(\mathop{\mathrm{Sz}}\nolimits(32))$
  • Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits(\mathop{\mathrm{Sz}}\nolimits(8))$
  • Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits(\mathop{\mathrm{PSL}}\nolimits(2,2^5))$
  • Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits(\mathop{\mathrm{Sz}}\nolimits(32))$
  • Figure : $\mathop{\mathrm{\overline{\Gamma}}}\nolimits(\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathrm{Sz}}\nolimits(32)))$
  • ...and 4 more figures

Theorems & Definitions (148)

  • Theorem 1
  • Definition 2
  • Remark 3
  • Lemma 4
  • Definition 5
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • ...and 138 more