[Paper Review] On conjugacy classes of SL$(2,q)$

Edith Adan-Bante, John M. Harris|LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas)|Apr 2, 2009

graph theory and CDMA systems4 references3 citations

TL;DR

This paper investigates the product of noncentral conjugacy classes in SL(2,q), proving that for even q, such products contain at least q−1 distinct conjugacy classes, while for odd q>3, they contain at least (q+3)/2 distinct classes. The results are sharp, with explicit constructions showing optimality, and extend earlier conjectures on products of conjugacy classes in finite simple groups.

ABSTRACT

Let SL(2,q) be the group of 2X2 matrices with determinant one over a finite field F of size q. We prove that if q is even, then the product of any two noncentral conjugacy classes of SL(2,q) is the union of at least q-1 distinct conjugacy classes of SL(2,q). On the other hand, if q>3 is odd, then the product of any two noncentral conjugacy classes of SL(2,q) is the union of at least (q+3)/2 distinct conjugacy classes of SL(2,q).

Motivation & Objective

• To determine the minimal number of distinct conjugacy classes in the product of two noncentral conjugacy classes in SL(2,q).
• To refine and extend Arad and Herzog's conjecture that products of nontrivial conjugacy classes in finite nonabelian simple groups are never single conjugacy classes.
• To establish sharp lower bounds on the size of such products, depending on the characteristic of the underlying finite field.
• To demonstrate the optimality of these bounds via explicit matrix constructions in SL(2,q) for both even and odd q.
• To analyze the structure of conjugacy classes in SL(2,q) using matrix representatives and trace-based invariants.

Proposed method

• Classify conjugacy classes in SL(2,q) into four types using matrix representatives over finite fields of size q.
• Use trace invariants and conjugation actions to analyze the structure of products of conjugacy classes.
• Employ trace computations such as Trace(A^{C(i)}B) = -i² + i(v−w) + w−2 to determine the number of distinct traces in products.
• Construct specific matrices C and D using field elements a,c and e,g to generate conjugates whose products yield matrices with controlled traces.
• Use properties of quadratic nonresidues and field automorphisms to show that the number of distinct traces in the product reaches the lower bound.
• Leverage group-theoretic lemmas (e.g., Lemma 2.15, Lemma 2.16) to prove that trace values are uniformly distributed over subsets of size (q+1)/2 or q−1, depending on parity.

Experimental results

Research questions

• RQ1 What is the minimal number of distinct conjugacy classes in the product of two noncentral conjugacy classes in SL(2,q) for even q?
• RQ2 How does this minimal number depend on the field size q when q is odd and greater than 3?
• RQ3 Can the lower bounds on the size of conjugacy class products be achieved, and if so, under what conditions?
• RQ4 Is the product of two noncentral conjugacy classes ever a single conjugacy class in SL(2,q)?
• RQ5 How do the algebraic properties of the finite field F_q, such as the existence of quadratic nonresidues, influence the structure of conjugacy class products?

Key findings

• For even q=2^m, the product of any two noncentral conjugacy classes in SL(2,q) contains at least q−1 distinct conjugacy classes.
• For odd q>3, the product of any two noncentral conjugacy classes in SL(2,q) contains at least (q+3)/2 distinct conjugacy classes.
• The bound of q−1 for even q is optimal, as demonstrated by explicit constructions where η(A^S B^S) = q−1.
• The bound of (q+3)/2 for odd q>3 is optimal, with explicit examples achieving equality, such as when A and B are unipotent matrices with non-square parameters.
• When q=3, the minimal product size is 2, which is less than (3+3)/2=3, showing that Theorem B does not extend to q=3.
• The results confirm that products of noncentral conjugacy classes in SL(2,q) are never single conjugacy classes when q≥4, supporting the Arad-Herzog conjecture.