Introduction to group theory. Isomorphism theorems. Permutation groups and Cayley's theorem. Conjugacy classes. Lagrange's theorem and the Sylow theorems. principle ideal domains. Polynomial ring.

### Algebra I (MATH 511)

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Programs\Type | Required | Core Elective | Area Elective |

Computer Science and Engineering - With Bachelor's Degree | * | ||

Computer Science and Engineering - With Master's Degree | * | ||

Computer Science and Engineering - With Thesis | * | ||

Cyber Security - With Bachelor's Degree | * | ||

Cyber Security - With Master's Degree | * | ||

Cyber Security - With Thesis | * | ||

Data Science - With Thesis | * | ||

Electronics Engineering and Computer Science - With Bachelor's Degree | * | ||

Electronics Engineering and Computer Science - With Master's Degree | * | ||

Electronics Engineering and Computer Science - With Thesis | * | ||

Electronics Engineering - With Bachelor's Degree | * | ||

Electronics Engineering - With Master's Degree | * | ||

Electronics Engineering - With Thesis | * | ||

Energy Technologies and Management-With Thesis | * | ||

Industrial Engineering - With Bachelor's Degree | * | ||

Industrial Engineering - With Master's Degree | * | ||

Industrial Engineering - With Thesis | * | ||

Leaders for Industry Biological Sciences and Bioengineering - Non Thesis | * | ||

Leaders for Industry Computer Science and Engineering - Non Thesis | * | ||

Leaders for Industry Electronics Engineering and Computer Science - Non Thesis | * | ||

Leaders for Industry Electronics Engineering - Non Thesis | * | ||

Leaders for Industry Industrial Engineering - Non Thesis | * | ||

Leaders for Industry Materials Science and Engineering - Non Thesis | * | ||

Leaders for Industry Mechatronics Engineering - Non Thesis | * | ||

Manufacturing Engineering - Non Thesis | * | ||

Manufacturing Engineering - With Bachelor's Degree | * | ||

Manufacturing Engineering - With Master's Degree | * | ||

Manufacturing Engineering - With Thesis | * | ||

Materials Science and Nano Engineering-(Pre:Materials Science and Engineering) | * | ||

Materials Science and Nano Engineering-(Pre:Materials Science and Engineering) | * | ||

Materials Science and Nano Engineering - With Thesis (Pre.Name: Materials Science and Engineering) | * | ||

Mathematics - With Bachelor's Degree | * | ||

Mathematics - With Master's Degree | * | ||

Mathematics - With Thesis | * | ||

Mechatronics Engineering - With Bachelor's Degree | * | ||

Mechatronics Engineering - With Master's Degree | * | ||

Mechatronics Engineering - With Thesis | * | ||

Molecular Biology, Genetics and Bioengineering (Prev. Name: Biological Sciences and Bioengineering) | * | ||

Molecular Biology, Genetics and Bioengineering-(Prev. Name: Biological Sciences and Bioengineering) | * | ||

Molecular Biology,Genetics and Bioengineering-With Thesis (Pre.Name:Biological Sciences and Bioeng.) | * | ||

Physics - Non Thesis | * | ||

Physics - With Bachelor's Degree | * | ||

Physics - With Master's Degree | * | ||

Physics - With Thesis | * |

### CONTENT

### OBJECTIVE

This is the first part of the two-semester basic algebra course for graduate students. The aim is to strengthen students' familiarity with basic algebraic structures which are commonly used in all parts of mathematics. These structures include groups, rings, vector spaces, modules and fields.

### LEARNING OUTCOMES

- Define and use Subgroups, Ideals, Homomorphisms and other basic concepts about Groups and Rings,
- Analyze and produce proofs of statements involving Groups and Rings,
- Use some fundamental examples such as the Symmetric Group and its Subgroups, Cyclic Groups, Matrix Groups, and Polynomial Rings,
- Distinguish basic Algebraic structures from each other up to Isomorphism,
- Use the three Isomorphism theorems for Groups and Rings,
- Use Direct and Semi-direct Product Constructions,
- Use Group Actions to study structures of Groups and to do effective counting in Groups,
- Apply Sylow Theorem and the Structure Theorem on Finitely Generated Abelian Groups,
- State basic Ring types such as Local Rings, Principal Ideal Domains, Unique Factorization Domains.

### Update Date:

### ASSESSMENT METHODS and CRITERIA

Percentage (%) | |

Final | 50 |

Midterm | 30 |

Participation | 10 |

Homework | 10 |

### RECOMENDED or REQUIRED READINGS

Textbook |
Hungerford, Thomas W. Algebra. Reprint of the 1974 original. Graduate Texts in Mathematics, 73. Springer-Verlag, New York-Berlin, 1980. xxiii+502 pp. ISBN: 0-387-90518-9. |

Readings |
Dummit, David S.; Foote, Richard M. Abstract algebra. Third edition. John Wiley \& Sons, Inc., Hoboken, NJ, 2004. xii+932 pp. ISBN: 0-471-43334-9. Lang, Serge Algebra. Revised third edition. Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 2002. xvi+914 pp. ISBN: 0-387-95385-X |