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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)].

##### MSC:
 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
##### Keywords:
finite simple groups; orders of elements