*Instructor: * Dr. Áron BERECZKY

*Text:* Peter J. Cameron: Introdction
to Algebra (Oxford Univerity Press) 1998. chapters 1, 2, 3,
sections 7.1.1, 7.1.2, 7.2.1.

*Prerequisite:* ---

*Course description:*

The course provides an
introduction to ring theory and group theory.

The methods
correspond to an introductory level.

*Topics:*
**Introduction:** relations, functions, operations,
polynomials, matrices.

**Elementary ring theory**:
rings, subrings, ideals, factor rings.

**Factorization
in rings:** 0-divisors, units, irreducibles, factorization,
Euclidean domains, PID, UFD and the connection between them.

**Fields:** maximal ideals in rings, quotient fields, field
of fractions, existence of simple extensions

**Elementary
group theory:** properties of groups, subgroups, cosets,
Lagrange's theorem, cyclic groups, order of an element.

**Homomorphisms: **Normal subgroups, factor groups,
isomorphism theorems, conjugacy.

**Group actions,
permutations: **Cayley's theorem, symmetric and alternating
groups, group actions and permutation groups, orbit, stabilizer,
groups of small order, symmetry groups, Sylow's theorems.