Minimal logarithmic signatures for finite groups of Lie type

Abstract

A logarithmic signature (LS) for a finite group G is an ordered tuple ? = [A 1, A 2, . . . , A n ] of subsets A i of G, such that every element $${g \in G}$$ can be expressed uniquely as a product g = a 1 a 2 . . . a n , where $${a_i \in A_i}$$ . The length of an LS ? is defined to be $${l(\alpha)= \sum^{n}_{i=1}|A_i|}$$ . It can be easily seen that for a group G of order $${\prod^k_{j=1}{p_j}^{m_j}}$$ , the length of any LS ? for G, satisfies, $${l(\alpha) \geq \sum^k_{j=1}m_jp_j}$$ . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS) (Gonzalez Vasco et al., Tatra Mt. Math. Publ. 25:2337, 2002). The MLS conjecture states that every finite simple group has an MLS. This paper addresses the MLS conjecture for classical groups of Lie type and is shown to be true for the families PSL n (q) and PSp 2n (q). Our methods use Singer subgroups and the Levi decomposition of parabolic subgroups for these groups.