04005nam 22005893 450 991091579430332120240723141346.01-4704-7543-X(MiAaPQ)EBC30671908(Au-PeEL)EBL30671908(PPN)27210678X(CKB)27902412200041(EXLCZ)992790241220004120230804d2023 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierAutomorphism Orbits and Element Orders in Finite Groups1st ed.Providence :American Mathematical Society,2023.©2023.1 online resource (108 pages)Memoirs of the American Mathematical Society Series ;v.287Print version: Bors, Alexander Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster Providence : American Mathematical Society,c2023 9781470465445 Cover -- Title page -- Chapter 1. Introduction -- 1.1. Statement of our main results -- 1.2. Overview of the proofs of Theorems 1.1.2 and 1.1.3 -- 1.3. Some related open questions -- Chapter 2. Notation -- Chapter 3. Proof of Theorem 1.1.3 -- 3.1. Sporadic groups -- 3.2. Alternating groups -- 3.3. Groups of Lie type -- Chapter 4. Proof of Theorem 1.1.2(1) -- Chapter 5. Proof of Theorem 1.1.2(2) -- 5.1. Reduction to semisimple groups -- 5.2. Two lemmas for working with partitions -- 5.3. Some auxiliary results on finite simple groups -- 5.4. Gaining some control over socle cosets in finite semisimple groups -- 5.5. Another equivalent reformulation of Theorem 1.1.2(2) -- 5.6. A bit of elementary number theory -- 5.7. Some results concerning the classes \Hcal_{ ̂, ̂, ̂} -- 5.8. More restrictions on finite semisimple groups with bounded \q-value -- 5.9. Completing the proof of Theorem 1.1.2(2) -- Bibliography -- Back Cover."For a finite group G, we denote by [omega](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) :[equals] [omega](G) - o(G) and q(G) :[equals] [omega](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 [omega](G) [equals] o(G)). We show that the index [absolute value]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 characterization of the Fischer-Griess Monster group M"--Provided by publisher.Memoirs of the American Mathematical Society SeriesFinite groupsAutomorphismsGroup theory and generalizations -- Abstract finite groups -- Arithmetic and combinatorial problemsmscGroup theory and generalizations -- Abstract finite groups -- Finite simple groups and their classificationmscGroup theory and generalizations -- Abstract finite groups -- AutomorphismsmscFinite groups.Automorphisms.Group theory and generalizations -- Abstract finite groups -- Arithmetic and combinatorial problems.Group theory and generalizations -- Abstract finite groups -- Finite simple groups and their classification.Group theory and generalizations -- Abstract finite groups -- Automorphisms.512/.23512.2320D6020D0520D45mscBors Alexander1779794Giudici Michael1779795Praeger Cheryl E1779796MiAaPQMiAaPQMiAaPQBOOK9910915794303321Automorphism Orbits and Element Orders in Finite Groups4303356UNINA