CaltechTHESIS
A Caltech Library Service

# Eigenvalue Structure in Primitive Linear Groups

## Citation

Huffman, William Cary (1974) Eigenvalue Structure in Primitive Linear Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1hnq-y922. https://resolver.caltech.edu/CaltechTHESIS:03092021-232801012

## Abstract

One approach to studying finite linear groups over the complex numbers is to classify those groups with an element possessing a certain eigenvalue structure. Let G be a finite group with a faithful, irreducible, primitive, unimodular complex representation X of degree n. Assume g ∈ G such that X(g) has eigenvalues ∈, ∈̅, 1, 1, . . ., 1 where ∈ is a primitive rth root of unity. H. F. Blichfeldt and J. H. Lindsey have classified G whenever r ⩾ 5. In this thesis r = 3 and 4 are handled. The main results are:

Theorem 1: Let G be a finite group with a faithful, irreducible, primitive, unimodular complex representation X of degree n. Assume there is an element g ∈ G such that X(g) has eigenvalues i, -i, 1, 1, . . ., 1. Then n ⩽ 4 and G is a known group.

Theorem 2: Let G be a finite group with a faithful, irreducible, primitive, unimodular complex representation X of degree n. Assume there is an element g ∈ G such that X(g) has eigenvalues ω, ω̅, 1, 1, . . ., 1 where ω = e2πi/3. Let N be the subgroup of G generated by all such elements. Then either

1. N ≅ An+1 and G/Z(G) ≅ An+1 or Sn+1·

2. n = 8, N = N', Z(N) has order 2, and N/Z(N) ≅ O8+(2); G/Z(G) is a subgroup of the automorphism group of O8+(2).

3. n ⩽ 7 and G is a known group.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
• Wales, David B.
Thesis Committee:
• Wales, David B.
Defense Date:14 September 1973
Funders:
Funding AgencyGrant Number
NSFUNSPECIFIED
Ford FoundationUNSPECIFIED
Record Number:CaltechTHESIS:03092021-232801012
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:03092021-232801012
DOI:10.7907/1hnq-y922
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14100
Collection:CaltechTHESIS
Deposited By: Benjamin Perez
Deposited On:11 Mar 2021 19:54