Groups of Prime Power Order Volume 1 (de Gruyter Expositions in Mathematics)

This 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. Machine Generated Contents Note: [§]93. Nonabelian 2-groups All Of Whose Minimal Nonabelian Subgroups Are Meta-cyclic And Have Exponent 4 -- [§]94. Nonabelian 2-groups All Of Whose Minimal Nonabelian Subgroups Are Non-metacyclic And Have Exponent 4 -- [§]95. Nonabelian 2-groups Of Exponent 2e Which Have No Minimal Nonabelian Subgroups Of Exponent 2e -- [§]96. Groups With At Most Two Conjugate Classes Of Nonnormal Subgroups -- [§]97.p-groups In Which Some Subgroups Are Generated By Elements Of Order P -- [§]98. Nonabelian 2-groups All Of Whose Minimal Nonabelian Subgroups Are Isomorphic To M2n+1, N [≥] 3 Fixed -- [§]99.2-groups With Sectional Rank At Most 4 -- [§]100.2-groups With Exactly One Maximal Subgroup Which Is Neither Abelian Nor Minimal Nonabelian -- [§]101.p-groups G With P> 2 And D(g) = 2 Having Exactly One Maximal Subgroup Which Is Neither Abelian Nor Minimal Nonabelian. Note Continued: [§]102.p-groups G With P> 2 And D(g)> 2 Having Exactly One Maximal Subgroup Which Is Neither Abelian Nor Minimal Nonabelian -- [§]103. Some Results Of Jonah And Konvisser -- [§]104. Degrees Of Irreducible Characters Of P-groups Associated With Finite Algebras -- [§]105. On Some Special P-groups -- [§]106. On Maximal Subgroups Of Two-generator 2-groups -- [§]107. Ranks Of Maximal Subgroups Of Nonmetacyclic Two-generator 2-groups -- [§]108.p-groups With Few Conjugate Classes Of Minimal Nonabelian Subgroups -- [§]109. On P-groups With Metacyclic Maximal Subgroup Without Cyclic Subgroup Of Index P -- [§]110. Equilibrated P-groups -- [§]111. Characterization Of Abelian And Minimal Nonabelian Groups -- [§]112. Non-dedekindian P-groups All Of Whose Nonnormal Subgroups Have The Same Order -- [§]113. The Class Of 2-groups In [§]70 Is Not Bounded. Note Continued: [§]114. Further Counting Theorems -- [§]115. Finite P-groups All Of Whose Maximal Subgroups Except One Are Extraspecial -- [§]116. Groups Covered By Few Proper Subgroups -- [§]117.2-groups All Of Whose Nonnormal Subgroups Are Either Cyclic Or Of Maximal Class -- [§]118. Review Of Characterizations Of P-groups With Various Minimal Nonabelian Subgroups -- [§]119. Review Of Characterizations Of P-groups Of Maximal Class -- [§]120. Nonabelian 2-groups Such That Any Two Distinct Minimal Nonabelian Subgroups Have Cyclic Intersection -- [§]121.p-groups Of Breadth 2 -- [§]122.p-groups All Of Whose Subgroups Have Normalizers Of Index At Most P -- [§]123. Subgroups Of Finite Groups Generated By All Elements In Two Shortest Conjugacy Classes -- [§]124. The Number Of Subgroups Of Given Order In A Metacyclic P-group. Note Continued: [§]125.p-groups G Containing A Maximal Subgroup H All Of Whose Subgroups Are G-invariant -- [§]126. The Existence Of P-groups G1