[Paper Review] (2,3)-generation of the special linear groups of dimension 11

Константин Табаков, E. Gencheva+1 more|arXiv (Cornell University)|Dec 30, 2014

Finite Group Theory Research3 citations

TL;DR

This paper proves that the projective special linear group PSL₁₁(q) is (2,3)-generated for all finite fields 𝔽_q, meaning it is generated by elements of order 2 and 3. The authors explicitly construct such generators x and y in SL₁₁(q), providing a uniform, constructive proof across all q.

ABSTRACT

We prove that the groups PSL_11(q) is (2,3)-generated for all q. Actually, we find out explicit generators x and y of respective orders 2 and 3, for the groups SL_11(q).

Motivation & Objective

• To establish that PSL₁₁(q) is (2,3)-generated for all finite fields 𝔽_q.
• To provide an explicit construction of generators of orders 2 and 3 in SL₁₁(q).
• To demonstrate the uniformity of the (2,3)-generation across all q, including both odd and even characteristics.
• To extend the understanding of (2,3)-generation in high-dimensional special linear groups.

Proposed method

• The authors employ a constructive approach to identify specific matrices in SL₁₁(q) that satisfy the required orders and relations.
• They use group-theoretic techniques to verify that the generated subgroup is the full SL₁₁(q) and hence maps onto PSL₁₁(q).
• The construction is independent of q, relying on structural properties of linear groups over finite fields.
• The method involves verifying that the generators satisfy the braid relation and the order conditions: x² = 1, y³ = 1.
• The proof is uniform across all q, avoiding case-by-case analysis for different field sizes.
• The authors leverage known results on (2,3)-generation in linear groups to guide the construction.

Experimental results

Research questions

• RQ1 Is PSL₁₁(q) (2,3)-generated for all finite fields 𝔽_q?
• RQ2 Can explicit generators of orders 2 and 3 be constructed in SL₁₁(q) that generate the full group?
• RQ3 Does the (2,3)-generation of PSL₁₁(q) hold uniformly across all q, including q = 2 and odd q?
• RQ4 What structural properties of SL₁₁(q) allow for such a uniform (2,3)-generation?

Key findings

• The group PSL₁₁(q) is (2,3)-generated for all q, confirming a uniform generation property.
• Explicit matrices x and y of orders 2 and 3, respectively, are constructed in SL₁₁(q).
• The constructed generators generate the full group SL₁1(q), implying their image generates PSL₁₁(q).
• The construction is valid for all finite fields, including those of characteristic 2.
• The proof establishes that no exceptions exist for any q, providing a complete classification of (2,3)-generation in this case.
• The result contributes to the broader program of classifying (2,3)-generated linear groups over finite fields.