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

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Abstract

For a finite group G, we denote by ω(G) the number ofAut(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) := ω(G) − o(G) and q(G) := ω(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 ω(G) = o(G)). We show that the index |G : Rad(G)| of the soluble radical Rad(G) of G can be bounded from above both by a function in d(G) and by a function in q(G) and o(Rad(G)). We also obtain a
curious quantitative characterisation of the Fischer-Griess Monster group M.
Original languageEnglish
Article number1427
Number of pages96
JournalMemoirs of the American Mathematical Society
Volume287
Issue number1426
DOIs
Publication statusPublished - 2023

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