Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster Jonathan E. Fielding and strong therapeutic practice

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Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster Jonathan E. Fielding and strong therapeutic practiceFor 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