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 |

MA-European Studies | |||

MA-European Studies-Non Thesis | |||

MA-Political Science | |||

MA-Political Science-Non Thes | |||

MA-Visual Arts&Vis. Com Des-NT | |||

MA-Visual Arts&Visual Com Des | |||

MS-Bio. Sci. & Bioeng. LFI | |||

MS-Bio. Sci. & Bioeng. LFI-ENG | |||

MS-Biological Sci&Bioeng. | * | ||

MS-Computer Sci.&Eng. LFI | |||

MS-Computer Sci.&Eng. LFI-ENG | |||

MS-Computer Science and Eng. | * | ||

MS-Cyber Security(with thesis) | * | ||

MS-Data Science | |||

MS-Elec. Eng&Comp Sc.LFI-ENG | |||

MS-Electronics Eng&Comp Sc.LFI | |||

MS-Electronics Eng&Computer Sc | * | ||

MS-Electronics Eng. | * | ||

MS-Electronics Eng. LFI | |||

MS-Electronics Eng. LFI-ENG | |||

MS-Energy Techno.&Man. | * | ||

MS-Industrial Eng. LFI-ENG | |||

MS-Industrial Engineering | * | ||

MS-Industrial Engineering LFI | |||

MS-Manufacturing Eng-Non Thes | * | ||

MS-Manufacturing Engineering | * | ||

MS-Materials Sci & Engineering | * | ||

MS-Materials Sci. & Eng. LFI | |||

MS-Materials Sci.&Eng. LFI-ENG | |||

MS-Mathematics | * | ||

MS-Mechatronics | * | ||

MS-Mechatronics LFI | |||

MS-Mechatronics LFI-ENG | |||

MS-Physics | |||

MS-Physics-Non Thesis | * | ||

MS-Psychology | |||

MS-Psychology-Non Thesis | |||

PHD-Biological Sci&Bioeng. | * | ||

PHD-Comp. Sci and Eng.after UG | * | ||

PHD-Computer Science and Eng. | * | ||

PHD-Cyber Security | * | ||

PHD-Electronics Eng&ComputerSc | * | ||

PHD-Electronics Eng. | * | ||

PHD-Electronics Eng. after UG | * | ||

PHD-Experimental Psychology | |||

PHD-Industrial Engineering | * | ||

PHD-Management | |||

PHD-Manufacturing Eng after UG | * | ||

PHD-Manufacturing Engineering | * | ||

PHD-Materials Sci.&Engineering | * | ||

PHD-Mathematics | |||

PHD-Mechatronics | * | ||

PHD-Mechatronics after UG | * | ||

PHD-Physics | |||

PHD-Physics after UG | |||

PHD-Social Psychology | |||

PHDBIO after UG | * | ||

PHDCYSEC after UG | * | ||

PHDEECS after UG | * | ||

PHDEPSY after UG | |||

PHDIE after UG | * | ||

PHDMAN after UG | |||

PHDMAN after UG-Finance | |||

PHDMAN after UG-Man. and Org. | |||

PHDMAN after UG-Op.&Sup. Cha. | |||

PHDMAN-Finance Area | |||

PHDMAN-Man. and Org. Area | |||

PHDMAN-Op. & Supp. Chain Area | |||

PHDMAT after UG | * | ||

PHDMATH after UG | * | ||

PHDSPSY after UG |

### 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 OUTCOME

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 |