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 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) ≅ O₈⁺(2); G/Z(G) is a subgroup of the automorphism group of O₈⁺(2).

3. n ⩽ 7 and G is a known group.

Thesis Files