Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster

For a finite group G G , we denote by ω ( G ) \omega (G) the number of A u t ( G ) Aut(G) -orbits on G G , and by o ( G ) o(G) the number of distinct element orders in G G . In this paper, we are primarily concerned with the two quantities d ( G ) ≔ ω ( G ) − o ( G ) \mathfrak {d}(G)≔\omega (G)-o(G) and q ( G ) ≔ ω ( G ) / o ( G ) \mathfrak {q}(G)≔\omega (G)/o(G) , each of which may be viewed as a measure for how far G G is from being an AT-group in the sense of Zhang (that is, a group with ω ( G ) = o ( G ) \omega (G)=o(G) ). We show that the index | G : R a d ( G ) | |G:Rad(G)| of the soluble radical R a d ( G ) Rad(G) of G G can be bounded from above both by a function in d ( G ) \mathfrak {d}(G) and by a function in q ( G ) \mathfrak {q}(G) and o ( R a d ( G ) ) o(Rad(G)) . We also obtain a curious quantitative characterisation of the Fischer-Griess Monster group M M .