LEADER 04005nam 22005893 450 001 9910915794303321 005 20240723141346.0 010 $a1-4704-7543-X 035 $a(MiAaPQ)EBC30671908 035 $a(Au-PeEL)EBL30671908 035 $a(PPN)27210678X 035 $a(CKB)27902412200041 035 $a(EXLCZ)9927902412200041 100 $a20230804d2023 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAutomorphism Orbits and Element Orders in Finite Groups 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2023. 210 4$dİ2023. 215 $a1 online resource (108 pages) 225 1 $aMemoirs of the American Mathematical Society Series ;$vv.287 311 08$aPrint version: Bors, Alexander Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster Providence : American Mathematical Society,c2023 9781470465445 327 $aCover -- 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. 330 $a"For a finite group G, we denote by [omega](G) the number of Aut(G)-orbits on G, and by o(G) the number of distinct element orders in G. In this paper, we are primarily concerned with the two quantities d(G) :[equals] [omega](G) - o(G) and q(G) :[equals] [omega](G)/ o(G), each of which may be viewed as a measure for how far G is from being an AT-group in the sense of Zhang (that is, a group with [omega](G) [equals] o(G)). We show that the index [absolute value]G : Rad(G) of the soluble radical Rad(G) of G can be bounded from above both by a function in d(G) and by a function in q(G) and o(Rad(G)). We also obtain a curious quantitative characterization of the Fischer-Griess Monster group M"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society Series 606 $aFinite groups 606 $aAutomorphisms 606 $aGroup theory and generalizations -- Abstract finite groups -- Arithmetic and combinatorial problems$2msc 606 $aGroup theory and generalizations -- Abstract finite groups -- Finite simple groups and their classification$2msc 606 $aGroup theory and generalizations -- Abstract finite groups -- Automorphisms$2msc 615 0$aFinite groups. 615 0$aAutomorphisms. 615 7$aGroup theory and generalizations -- Abstract finite groups -- Arithmetic and combinatorial problems. 615 7$aGroup theory and generalizations -- Abstract finite groups -- Finite simple groups and their classification. 615 7$aGroup theory and generalizations -- Abstract finite groups -- Automorphisms. 676 $a512/.23 676 $a512.23 686 $a20D60$a20D05$a20D45$2msc 700 $aBors$b Alexander$01779794 701 $aGiudici$b Michael$01779795 701 $aPraeger$b Cheryl E$01779796 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910915794303321 996 $aAutomorphism Orbits and Element Orders in Finite Groups$94303356 997 $aUNINA