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[null,"1.","Record Nr.",null,"UNINA9910915794303321"]
[null,null,"Autore",null,"Bors Alexander"]
[null,null,"Titolo",null,"Automorphism Orbits and Element Orders in Finite Groups"]
[null,null,"Pubbl/distr/stampa",null,"Providence : , : American Mathematical Society, , 2023"]
[{"$nulls":4},"©2023"]
[null,null,"ISBN",null,"1-4704-7543-X"]
[null,null,"Edizione",null,"[1st ed.]"]
[null,null,"Descrizione fisica",null,"1 online resource (108 pages)"]
[null,null,"Collana",null,"Memoirs of the American Mathematical Society Series ; ; v.287"]
[null,null,"Classificazione",null,"20D6020D0520D45"]
[null,null,"Disciplina",null,"512/.23"]
[{"$nulls":4},"512.23"]
[null,null,"Soggetti",null,"Finite groups"]
[{"$nulls":4},"Automorphisms"]
[{"$nulls":4},"Group theory and generalizations -- Abstract finite groups -- Arithmetic and combinatorial problems"]
[{"$nulls":4},"Group theory and generalizations -- Abstract finite groups -- Finite simple groups and their classification"]
[{"$nulls":4},"Group theory and generalizations -- Abstract finite groups -- Automorphisms"]
[null,null,"Lingua di pubblicazione",null,"Inglese"]
[null,null,"Formato",null,"Materiale a stampa"]
[null,null,null,"Livello bibliografico","Monografia"]
[null,null,"Nota di contenuto",null,"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."]
[null,null,null,"Sommario/riassunto","\"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 "]
[{"$nulls":4},"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\"--"]