*Group Theory* is intended as a textbook for a one-term course in group theory for senior undergraduate or graduate students. It provides students with good insight into group theory as quickly as possible. Simplicity and working knowledge are emphasized here over mathematical completeness. This book will provide a rigorous proof-based modern treatment of the main results of group theory. As a result, proofs are in great details with a lot of interesting examples.

This book can also serve as a reference for professional mathematicians. The book contains 221 carefully selected exercises of varying difficulty which will allow students to practice their own computational and proof-writing skills. Sample solutions to some exercise questions are provided, from which students can learn to write their own solutions and proofs. Besides standard ones, many of the exercises are very interesting.

*Group Theory *includes:

- basic properties of groups
- symmetric groups and alternating groups
- isomorphism theorems, commutator subgroups, direct and semi-direct products of groups
- group action on sets, Cauchy-Frobenius Lemma, symmetries of the regular polyhedrons
- Jordan-Holder Theorem and Sylow Theorems
- free groups, group presentations, finitely generated abelian groups
- solvable groups, Hall subgroups, characteristically simple groups, Sylow systems and Sylow bases
- nilpotent groups, Frattini subgroups
- basic properties of representations of groups on vector spaces, subrepresentations; irreducibility, Schur's Lemma, and Maschke's Theorem

**Basic Theory on Groups**

1.1 Basic properties of groups

1.2 Groups of permutations

1.3 Isomorphism theorems

1.4 Commutator subgroups and dihedral groups

1.5 Exercises

**2. Group Actions**

2.1 Group action on a set

2.2 Applications of -sets to counting

2.3 Symmetries of the regular polyhedrons

2.4 Simplegroups

2.5 Exercises

**3. Jordan–Holder Theorem and Sylow Theorems**

3.1 Series of groups

3.2 Composition series of groups

3.3 Finite –groups

3.4 Sylowtheorems

3.5 Applications of the Sylow Theorems

3.6 Exercises

**4. Free Groups**

4.1 Free abelian groups

4.2 Freegroups

4.3 Group presentations

4.4 Exercises

**5. Solvable Groups**

5.1 Basic properties of solvable groups

5.2 Finite solvable groups

5.3 Hall subgroups

5.4 Sylow systems and Sylow bases

5.5 Exercises

**6. Nilpotent Groups**

6.1 Basic properties of nilpotent groups

6.2 Frattini subgroups

6.3 Exercises

**7. Representations of Groups**

7.1 Definitions and examples

7.2 Subrepresentations, quotient representations, and homomorphisms

7.3 The methods of constructing representations

7.4 Irreducible representations, completely reducible representations

7.5 Maschke’s Theorem

7.6 Exercises

**Sample Solutions**

**Appendix A: EquivalenceRelations andKuratowski–Zorn Lemma**

**References**

**Index**