We prove that the order of any subloop of a finite Moufang loop is a factor of the order of the loop, thus obtaining an analog of Lagrange's theorem for finite Moufang loops.

We obtain the first example of an infinite series of finite simple groups that are uniquely determined by their prime graph in the class of all finite groups. We also show that there exist almost simple groups for which the number of finite groups with the same prime graph is equal to 2.

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We show that if a finite simple group G, isomorphic to PSL n (q) or PSU n (q) where either n = 4 or q is prime or even, acts on a vector space over a field of the defining characteristic of G; then the corresponding semidirect product contains an element whose order is distinct from every element order of G. We infer that the group PSL n (q), n = 4 or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka Notebook.

Groups with triality, which arose in the papers of Glauberman and Doro, are naturally connected with Moufang loops. In this paper, we describe all possible, in a sense, groups with triality associated with a given Moufang loop. We also introduce several universal groups with triality and discuss their properties.