The Existence of Minimal Logarithmic Signatures for Some Finite Simple Groups

摘要

A logarithmic signature for a finite group G is a sequence = [A(1), ..., A(s)] of subsets of G such that every element g G can be uniquely written in the form g = g(1)...g(s), where g(i) A(i), 1 i s. The number Sigma(s)(i = 1)|A(i)| is called the length of and denoted by l(). A logarithmic signature is said to be minimal (MLS) if l() = Sigma(n)(i = 1)m(i)p(i), where is the prime factorization of |G|. The MLS conjecture states that every finite simple group has an MLS. The aim of this article is proving the existence of a minimal logarithmic signature for the untwisted groups G(2)(3(n)), the orthogonal groups (7)(q) and P-8(+)(q), q is an odd prime power, the orthogonal groups (9)(3), P-10(+)(3), and P-8(-)(3), the Tits simple group F-2(4)(2), the Janko group J(3), the twisted group D-3(4)(2), the Rudvalis group Ru, and the Fischer group Fi(22). As a consequence of our results, it is proved that all finite groups of order 10(12) other than the Ree group Ree(27), the O'Nan group ON, and the untwisted group G(2)(7) have MLS.