Group theory
著者
書誌事項
Group theory
Springer, c2023
大学図書館所蔵 全2件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
注記
Includes index
内容説明・目次
内容説明
This textbook focuses on the basics and complex themes of group theory taught to senior undergraduate mathematics students across universities. The contents focus on the properties of groups, subgroups, cyclic groups, permutation groups, cosets and Lagrange's theorem, normal subgroups and factor groups, group homomorphisms and isomorphisms, automorphisms, direct products, group actions and Sylow theorems. Pedagogical elements such as end of chapter exercises and solved problems are included to help understand abstract notions. Intermediate lemmas are also carefully designed so that they not only serve the theorems but are also valuable independently. The book is a useful reference to undergraduate and graduate students besides academics.
目次
1. Group....................................................................................................... 1-58
1.1 Groups................................................................................................... 4
1.2 Cayley Table.......................................................................................... 8
1.3 Elementary Properties of Groups........................................................ 32
1.4 Dihedral Groups.................................................................................. 49
2. Finite Groups and Subgroups.............................................................. 59-98
2.1 Finite Groups....................................................................................... 59
2.2 Subgroups............................................................................................ 70
2.3 Subgroup Tests.................................................................................... 71
2.4 Special Class of Subgroups................................................................. 82
2.5 Intersection and Union of Subgroups................................................. 91
2.6 Product of Two Subgroups................................................................ 93
3. Cyclic Groups..................................................................................... 99-118
3.1 Cyclic Groups and their Properties..................................................... 99
3.2 Generators of a Cyclic Group........................................................... 102
3.3 Subgroups of Cyclic Groups............................................................. 104
4. Permutation Groups......................................................................... 119-142
4.1 Permutation of a Set.......................................................................... 119
4.2 Permutation Group of a Set.............................................................. 121
4.3 Cycle Notation................................................................................... 124
4.4 Theorems on Permutations and Cycles .......................................... 126
4.5 Even and Odd Permutations.............................................................. 134
4.6 Alternating Group of Degree n......................................................... 138
5. Cosets and Lagrange's Theorem................................................... 143-168
5.1 Definition of Cosets and Properties of Cosets.................................. 143
5.2 Lagrange's Theorem and its Applications........................................ 148
5.3 Application of Cosets to Permutation Groups.................................. 164
(xii)
6. Normal Subgroups and Factor Groups ........................................ 169-194
6.1 Normal Subgroup and Equivalent Conditions for a Subgroup to be
Normal............................................................................................... 169
6.2 Factor Groups.................................................................................... 180
6.3 Commutator Subgroup of a Group and its Properties...................... 187
6.4 The G/Z Theorem.............................................................................. 189
6.5 Cauchy's Theorem for Abelian Group............................................. 191
7. Group Homomorphism and Isomorphism........................................ 195-222
7.1 Homomorphism of Groups and its Properties.................................. 195
7.2 Properties of Subgroups under Homomorphism............................... 200
7.3 Isomorphism of Groups..................................................................... 205
7.4 Some Theorems Based on Isomorphism of Groups......................... 207
8. Automorphisms ................................................................................. 223-240
8.1 Automorphism of a Group................................................................ 223
8.2 Inner Automorphisms........................................................................ 226
8.3 Theorems Based on Automorphism of a Group............................... 228
9. Direct Products............................................................................... 241-270
9.1 External Direct Product..................................................................... 241
9.2 Properties of External Direct Products............................................. 244
9.3 U(n) as External Direct Products...................................................... 249
9.4 Internal Direct Products..................................................................... 254
9.5 Fundamental Theorem of Finite Abelian Groups............................. 258
10. Group Actions.................................................................................. 271-302
10.1 Group Actions................................................................................. 271
10.2 Kernels, Orbits and Stabilizers........................................................ 275
10.3 Group acting on themselves by Conjugation.................................. 291
10.4 Conjugacy in Sn.............................................................................. 296
11. Sylow Theorems............................................................................... 303-325
11.1 p-Groups and Sylow p-subgroups.................................................. 303
11.2 Simple Groups................................................................................. 309
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