LEADER 06396nam 2200673Ia 450 001 9910454598003321 005 20200520144314.0 010 $a1-281-99347-6 010 $a9786611993474 010 $a3-11-020822-9 024 7 $a10.1515/9783110208221 035 $a(CKB)1000000000698066 035 $a(EBL)429279 035 $a(OCoLC)808801315 035 $a(SSID)ssj0000165945 035 $a(PQKBManifestationID)11164401 035 $a(PQKBTitleCode)TC0000165945 035 $a(PQKBWorkID)10147042 035 $a(PQKB)11220398 035 $a(MiAaPQ)EBC429279 035 $a(DE-B1597)34844 035 $a(OCoLC)979954977 035 $a(DE-B1597)9783110208221 035 $a(PPN)175523851$9sudoc 035 $a(PPN)140858237 035 $a(Au-PeEL)EBL429279 035 $a(CaPaEBR)ebr10275903 035 $a(CaONFJC)MIL199347 035 $a(EXLCZ)991000000000698066 100 $a20081001d2008 uy 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt 182 $cc 183 $acr 200 10$aGroups of prime power order$hVolume 1$b[electronic resource] /$fby Yakov Berkovich 210 $aBerlin ;$aNew York $cW. de Gruyter$dc2008 215 $a1 online resource (532 p.) 225 1 $aDe Gruyter expositions in mathematics,$x0938-6572 ;$v46 300 $aDescription based upon print version of record. 311 $a3-11-020418-5 320 $aIncludes bibliographical references and indexes. 327 $t Frontmatter -- $tContents -- $tList of definitions and notations -- $tForeword -- $tPreface -- $tIntroduction -- $t§1. Groups with a cyclic subgroup of index p. Frattini subgroup. Varia -- $t§2. The class number, character degrees -- $t§3. Minimal classes -- $t§4. p-groups with cyclic Frattini subgroup -- $t§5. Hall's enumeration principle -- $t§6. q'-automorphisms of q-groups -- $t§7. Regular p-groups -- $t§8. Pyramidal p-groups -- $t§9. On p-groups of maximal class -- $t§10. On abelian subgroups of p-groups -- $t§11. On the power structure of a p-group -- $t§12. Counting theorems for p-groups of maximal class -- $t§13. Further counting theorems -- $t§14. Thompson's critical subgroup -- $t§15. Generators of p-groups -- $t§16. Classification of finite p-groups all of whose noncyclic subgroups are normal -- $t§17. Counting theorems for regular p-groups -- $t§18. Counting theorems for irregular p-groups -- $t§19. Some additional counting theorems -- $t§20. Groups with small abelian subgroups and partitions -- $t§21. On the Schur multiplier and the commutator subgroup -- $t§22. On characters of p-groups -- $t§23. On subgroups of given exponent -- $t§24. Hall's theorem on normal subgroups of given exponent -- $t§25. On the lattice of subgroups of a group -- $t§26. Powerful p-groups -- $t§27. p-groups with normal centralizers of all elements -- $t§28. p-groups with a uniqueness condition for nonnormal subgroups -- $t§29. On isoclinism -- $t§30. On p-groups with few nonabelian subgroups of order pp and exponent p -- $t§31. On p-groups with small p0-groups of operators -- $t§32. W. Gaschütz's and P. Schmid's theorems on p-automorphisms of p-groups -- $t§33. Groups of order pm with automorphisms of order pm-1, pm-2 or pm-3 -- $t§34. Nilpotent groups of automorphisms -- $t§35. Maximal abelian subgroups of p-groups -- $t§36. Short proofs of some basic characterization theorems of finite p-group theory -- $t§37. MacWilliams' theorem -- $t§38. p-groups with exactly two conjugate classes of subgroups of small orders and exponentp > 2 -- $t§39. Alperin's problem on abelian subgroups of small index -- $t§40. On breadth and class number of p-groups -- $t§41. Groups in which every two noncyclic subgroups of the same order have the same rank -- $t§42. On intersections of some subgroups -- $t§43. On 2-groups with few cyclic subgroups of given order -- $t§44. Some characterizations of metacyclic p-groups -- $t§45. A counting theorem for p-groups of odd order -- $tAppendix 1. The Hall-Petrescu formula -- $tAppendix 2. Mann's proof of monomiality of p-groups -- $tAppendix 3. Theorems of Isaacs on actions of groups -- $tAppendix 4. Freiman's number-theoretical theorems -- $tAppendix 5. Another proof of Theorem 5.4 -- $tAppendix 6. On the order of p-groups of given derived length -- $tAppendix 7. Relative indices of elements of p-groups -- $tAppendix 8. p-groups withabsolutely regular Frattini subgroup -- $tAppendix 9. On characteristic subgroups of metacyclic groups -- $tAppendix 10. On minimal characters of p-groups -- $tAppendix 11. On sums of degrees of irreducible characters -- $tAppendix 12. 2-groups whose maximal cyclic subgroups of order > 2 are self-centralizing -- $tAppendix 13. Normalizers of Sylow p-subgroups of symmetric groups -- $tAppendix 14. 2-groups with an involution contained in only one subgroup of order 4 -- $tAppendix 15. A criterion for a group to be nilpotent -- $tResearch problems and themes I -- $t Backmatter 330 $aThis is the first of three volumes of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this monograph include: (a) counting of subgroups, with almost all main counting theorems being proved, (b) regular p-groups and regularity criteria, (c) p-groups of maximal class and their numerous characterizations, (d) characters of p-groups, (e) p-groups with large Schur multiplier and commutator subgroups, (f) (p-1)-admissible Hall chains in normal subgroups, (g) powerful p-groups, (h) automorphisms of p-groups, (i) p-groups all of whose nonnormal subgroups are cyclic, (j) Alperin's problem on abelian subgroups of small index. The book is suitable for researchers and graduate students of mathematics with a modest background on algebra. It also contains hundreds of original exercises (with difficult exercises being solved) and a comprehensive list of about 700 open problems. 410 0$aGruyter expositions in mathematics ;$v46. 606 $aFinite groups 606 $aGroup theory 608 $aElectronic books. 615 0$aFinite groups. 615 0$aGroup theory. 676 $a512.23 700 $aBerkovich$b Yakov$0472302 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910454598003321 996 $aGroups of prime power order$9230033 997 $aUNINA