A characteristic property for each finite projective special linear group. (English) Zbl 0718.20009

Groups, Sel. Pap. Aust. Natl. Univ. Group Theory Program, 3rd Int. Conf. Theory Groups Rel. Top., Canberra/Aust. 1989, Lect. Notes Math. 1456, 171-180 (1990).
[For the entire collection see Zbl 0706.00012.]
In [Group Theory, Proc. Conf. Singapore 1987, 531-540 (1989; Zbl 0657.20017)] the first author made the following conjecture: Let \(G\) be a group and \(M\) a finite simple group with (1) \(\pi_ e(G)=\pi_ e(M)\), (2) \(|G|=|M|\), then \(G\simeq M\) (\(\pi_ e(X)\) denotes the set of orders of elements in the finite group \(X\)).
Here this conjecture is verified for the groups \(M\simeq L_ 2(q)\), \(L_ 3(q)\) and \(L_ 4(3)\), \(L_ 5(3)\), \(L_ 4(2),...,L_{10}(2)\). This paper follows similar ones of the first author [Contemp. Math. 82, 171-180 (1989; Zbl 0668.20019), and J. Math., Wuhan Univ. 5, 191-200 (1985; Zbl 0597.20007)].


20D05 Finite simple groups and their classification
20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups