# MA136 content

Content:

#### Section 1 Group Theory:

• Motivating examples: numbers, symmetry groups
• Definitions, elementary properties
• Subgroups, including subgroups of \$Z\$
• Arithmetic modulo n and the group \$Z_n\$
• Lagrange's Theorem
• Permutation groups, odd and even permutations (proof optional)
• Normal subgroups and quotient groups

#### Section 2 Ring Theory:

• Definitions: Commutative and non-commutative rings, integral domains, fields
• Examples: \$Z, Q, R, C, Z_n\$, matrices, polynomials, Gaussian integers

Aims:

To introduce First Year Mathematics students to abstract Algebra, covering Group Theory and Ring Theory.

Objectives:

By the end of the module students should be able to understand:

• the abstract definition of a group, and be familiar with the basic types of examples, including numbers, symmetry groups and groups of permutations and matrices.
• what subgroups are, and be familiar with the proof of Lagrange’s Theorem.
• the definition of various types of ring, and be familiar with a number of examples, including numbers, polynomials, and matrices.
• unit groups of rings, and be able to calculate the unit groups of the integers modulo n.

Books:

Any library book with Abstract Algebra in the title would be useful.