Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster Jonathan E. Fielding The African Renaissance and the

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Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster Jonathan E. Fielding The African Renaissance and theFor a finite group G, we denote by ?(G) the number of Aut(G) orbits on G, and by o(G) the number of distinct element orders in G. In this paper, we are primarily concerned with the two quantities d(G) := ?(G) ? o(G) and q(G) := ?(G) o(G), each of which may be viewed as a measure for how far G is from being an AT group in the sense of Zhang (that is, a group with ?(G) = o(G)). We show that the index G : Rad(G) of the soluble radical Rad(G) of G can be