LEADER 03819nam 22004935 450 001 9910483745203321 005 20251113181106.0 010 $a3-030-74100-1 024 7 $a10.1007/978-3-030-74100-6 035 $a(CKB)4100000011946204 035 $a(MiAaPQ)EBC6631316 035 $a(Au-PeEL)EBL6631316 035 $a(OCoLC)1253354336 035 $a(PPN)255881347 035 $a(DE-He213)978-3-030-74100-6 035 $a(EXLCZ)994100000011946204 100 $a20210525d2021 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 14$aThe Spread of Almost Simple Classical Groups /$fby Scott Harper 205 $a1st ed. 2021. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2021. 215 $a1 online resource (158 pages) 225 1 $aLecture Notes in Mathematics,$x1617-9692 ;$v2286 311 08$a3-030-74099-4 327 $aIntro -- Preface -- Contents -- 1 Introduction -- 2 Preliminaries -- Notational Conventions -- 2.1 Probabilistic Method -- 2.2 Classical Groups -- 2.3 Actions of Classical Groups -- 2.4 Standard Bases -- 2.5 Classical Algebraic Groups -- 2.6 Maximal Subgroups of Classical Groups -- 2.7 Computational Methods -- 3 Shintani Descent -- 3.1 Introduction -- 3.2 Properties -- 3.3 Applications -- 3.4 Generalisation -- 4 Fixed Point Ratios -- 4.1 Subspace Actions -- 4.2 Nonsubspace Actions -- 5 Orthogonal Groups -- 5.1 Introduction -- 5.2 Automorphisms -- 5.2.1 Preliminaries -- 5.2.2 Plus-Type -- 5.2.3 Minus-Type -- 5.2.4 Conjugacy of Outer Automorphisms -- 5.3 Elements -- 5.3.1 Preliminaries -- 5.3.2 Types of Semisimple Elements -- 5.3.3 Reflections -- 5.3.4 Field Extension Subgroups -- 5.4 Case I: Semilinear Automorphisms -- 5.4.1 Case I(a) -- 5.4.2 Case I(b) -- 5.5 Case II: Linear Automorphisms -- 5.5.1 Case II(a) -- 5.5.2 Case II(b) -- 5.6 Case III: Triality Automorphisms -- 5.6.1 Case III(a) -- 5.6.2 Case III(b) -- 5.6.3 Case III(c) -- 6 Unitary Groups -- 6.1 Introduction -- 6.2 Automorphisms -- 6.3 Elements -- 6.4 Case I: Semilinear Automorphisms -- 6.4.1 Case I(a) -- 6.4.2 Case I(b) -- 6.5 Case II: Linear Automorphisms -- 6.5.1 Case II(a) -- 6.5.2 Case II(b) -- 6.6 Linear Groups -- A Magma Code -- References. 330 $aThis monograph studies generating sets of almost simple classical groups, by bounding the spread of these groups. Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group, every nontrivial element belongs to a generating pair. Groups with this property are said to be 3/2-generated. Breuer, Guralnick and Kantor conjectured that a finite group is 3/2-generated if and only if every proper quotient is cyclic. We prove a strong version of this conjecture for almost simple classical groups, by bounding the spread of these groups. This involves analysing the automorphisms, fixed point ratios and subgroup structure of almost simple classical groups, so the first half of this monograph is dedicated to these general topics. In particular, we give a general exposition of Shintani descent. This monograph will interest researchers in group generation, but the opening chapters also serve as a general introduction to the almost simple classical groups. . 410 0$aLecture Notes in Mathematics,$x1617-9692 ;$v2286 606 $aGroup theory 606 $aGroup Theory and Generalizations 615 0$aGroup theory. 615 14$aGroup Theory and Generalizations. 676 $a512.2 700 $aHarper$b Scott$c(Mathematician),$0849388 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910483745203321 996 $aThe Spread of Almost Simple Classical Groups$91896891 997 $aUNINA