Kleshchev Algebra
Student Solution Manual
Chapter 1 through 5
James Wilson
1 Groups
• Sylow Theorems
• Simple Groups
• Chain Conditions
2 Fields
• Galois Theory
• Finite Fields
• Transcendental Extensions
3 Modules
• Over PIDs
• Semi-simplicity
• Algebras
4 Categories
• Commutative Diagrams
• Naturality
• Category Equivalence
5 Commutative Algebra
• Localization/Extensions
• Radicals
• Algebraic Geometry
,c 2002-2003. James Wilson, University of Oregon.
◯
Department of Mathematics
1222 University of Oregon
Eugene, OR 97403-1222
www.uoregon.edu/ jwilson7/
Written with L AT E X 2ε.
Please Recycle when finished.
This is the product of a graduate level algebra course taken at the University
of Oregon, 2002-2003, under the instruction of Alexander Kleshchev. The so-
lutions are mostly the work of the author, James Wilson, while the exercises
were developed by Kleshchev. Many exercises appear in common texts, such as
Hungerford’s Algebra and Rotman’s Advanced Modern Algebra.
I developed the solutions as a preparation for the Ph.D. qualifying exams. With
the exception of some parts of chapter 2, most of the solutions have been proof
read to some degree. However, mistakes are bound to exist still. My hope is
that these may help others with many of the common problems encountered
when preparing for qualifying exams. Having passed the exams, my advice in
using these notes is to do each problem yourself.
I welcome any comments and corrections you may have. I would also like to
thank Professor Kleshchev, David Hill, Alex Jordan, and Dragos Neacsu for
their advice on certain solutions.
,
, Contents
1 Groups 5
2 Fields 35
3 Modules 65
4 Categories 125
5 Rings 127
Chapter 1
Groups
1 T/F Cyclic Groups ......................................................................... 7
2 Transitive Embedding............................................................ 7
3 Probability of Commutativity ............................................... 7
4 Infinite Groups ....................................................................... 8
5 Frattini Subgroup .................................................................. 8
6 T/F Cyclic Automorphisms .......................................................... 8
7 Maximal p-subgroups ............................................................ 9
8 Minimal p-quotients .............................................................. 9
9 T/F Nilpotent Subgroups ............................................................ 10
10 Classification of “Local” Groups ........................................ 10
11 p-subgroup Chains................................................................ 11
12 T/F Counting Involutions ........................................................... 11
13 Nilpotent Extensions ........................................................... 12
14 T/F Normal Transitivity.............................................................. 12
15 Examples of the Parallelogram Law .................................. 12
16 The Complex Subgroup ...................................................... 13
17 Parallelogram Law ............................................................... 13
18 HK-subgroup ..................................................................... 13
19 T/F Hamiltonian Groups ............................................................ 14
20 Center of Sn ........................................................................................................ 14
21 Parallelogram Example........................................................ 14
22 Q subgroups ......................................................................... 15
23 Finite Index Intersections ................................................... 15
24 Unions of Conjugation ......................................................... 15
25 Unions of Conjugation ......................................................... 16
26 Conjugacy Classes and Generators .................................... 16
27 GLn(Fq) ................................................................................. 16
Student Solution Manual
Chapter 1 through 5
James Wilson
1 Groups
• Sylow Theorems
• Simple Groups
• Chain Conditions
2 Fields
• Galois Theory
• Finite Fields
• Transcendental Extensions
3 Modules
• Over PIDs
• Semi-simplicity
• Algebras
4 Categories
• Commutative Diagrams
• Naturality
• Category Equivalence
5 Commutative Algebra
• Localization/Extensions
• Radicals
• Algebraic Geometry
,c 2002-2003. James Wilson, University of Oregon.
◯
Department of Mathematics
1222 University of Oregon
Eugene, OR 97403-1222
www.uoregon.edu/ jwilson7/
Written with L AT E X 2ε.
Please Recycle when finished.
This is the product of a graduate level algebra course taken at the University
of Oregon, 2002-2003, under the instruction of Alexander Kleshchev. The so-
lutions are mostly the work of the author, James Wilson, while the exercises
were developed by Kleshchev. Many exercises appear in common texts, such as
Hungerford’s Algebra and Rotman’s Advanced Modern Algebra.
I developed the solutions as a preparation for the Ph.D. qualifying exams. With
the exception of some parts of chapter 2, most of the solutions have been proof
read to some degree. However, mistakes are bound to exist still. My hope is
that these may help others with many of the common problems encountered
when preparing for qualifying exams. Having passed the exams, my advice in
using these notes is to do each problem yourself.
I welcome any comments and corrections you may have. I would also like to
thank Professor Kleshchev, David Hill, Alex Jordan, and Dragos Neacsu for
their advice on certain solutions.
,
, Contents
1 Groups 5
2 Fields 35
3 Modules 65
4 Categories 125
5 Rings 127
Chapter 1
Groups
1 T/F Cyclic Groups ......................................................................... 7
2 Transitive Embedding............................................................ 7
3 Probability of Commutativity ............................................... 7
4 Infinite Groups ....................................................................... 8
5 Frattini Subgroup .................................................................. 8
6 T/F Cyclic Automorphisms .......................................................... 8
7 Maximal p-subgroups ............................................................ 9
8 Minimal p-quotients .............................................................. 9
9 T/F Nilpotent Subgroups ............................................................ 10
10 Classification of “Local” Groups ........................................ 10
11 p-subgroup Chains................................................................ 11
12 T/F Counting Involutions ........................................................... 11
13 Nilpotent Extensions ........................................................... 12
14 T/F Normal Transitivity.............................................................. 12
15 Examples of the Parallelogram Law .................................. 12
16 The Complex Subgroup ...................................................... 13
17 Parallelogram Law ............................................................... 13
18 HK-subgroup ..................................................................... 13
19 T/F Hamiltonian Groups ............................................................ 14
20 Center of Sn ........................................................................................................ 14
21 Parallelogram Example........................................................ 14
22 Q subgroups ......................................................................... 15
23 Finite Index Intersections ................................................... 15
24 Unions of Conjugation ......................................................... 15
25 Unions of Conjugation ......................................................... 16
26 Conjugacy Classes and Generators .................................... 16
27 GLn(Fq) ................................................................................. 16
