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Record Nr. |
UNINA9910915794303321 |
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Autore |
Bors Alexander |
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Titolo |
Automorphism Orbits and Element Orders in Finite Groups |
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Pubbl/distr/stampa |
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Providence : , : American Mathematical Society, , 2023 |
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©2023 |
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ISBN |
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Edizione |
[1st ed.] |
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Descrizione fisica |
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1 online resource (108 pages) |
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Collana |
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Memoirs of the American Mathematical Society Series ; ; v.287 |
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Classificazione |
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Altri autori (Persone) |
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GiudiciMichael |
PraegerCheryl E |
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Disciplina |
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Soggetti |
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Finite 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 |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di contenuto |
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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. |
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Sommario/riassunto |
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"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"-- |
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