## 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 r^{th} 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 = e^{2πi/3}. Let N be the subgroup of G generated by all such elements. Then either

1. N ≅ A_{n+1} and G/Z(G) ≅ A_{n+1} or S_{n+1}·

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

3. n ⩽ 7 and G is a known group.

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||||
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Subject Keywords: | Mathematics | ||||||

Degree Grantor: | California Institute of Technology | ||||||

Division: | Physics, Mathematics and Astronomy | ||||||

Major Option: | Mathematics | ||||||

Thesis Availability: | Public (worldwide access) | ||||||

Research Advisor(s): | - Wales, David B.
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Thesis Committee: | - Wales, David B.
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Defense Date: | 14 September 1973 | ||||||

Funders: |
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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 | ||||||

Last Modified: | 11 Mar 2021 19:55 |

## Thesis Files

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