28357 - Algebra 1

Learning outcomes

At the end of the lectures the student will have the basic knowledge of abstract algebra: s/he will be aware of a rigorous definition natural, integer and rational numbers and s/he will know the basic algebraic structures such as posets, groups. The student can take advantage of this knowledge for mastering a mathematical reasoning.

Course contents

Elementary theory of sets: inclusion, union, intersection, complement, difference. The set of subsets, De Morgan laws, Cartesian product. Relations. Functions.

Relations in AxA: reflexive, symmetric, antisymmetric, transitive. Functions: definitions and terminology. Injective, surjective and bijective functions. 

Basic elements in cardinality theory.

Peano axioms. Induction. Basic elements in enumerative combinatorics.

Partitions and equivalence relations, Quotient set. Example: congruence modulo n.

Construction of integer and rational numbers as equivalence classes

Order relations and posets.

Divisibility for integer numbers: division lemma and GCD.

Euclid's algorithm, Bézout identity. Prime numbers. Infinity of prime numbers.

Fundamental theorem of arithmetic

Groups. Abelian groups. The general linear group and the symmetric group. Cancellation law. Multiplication tables.

Example of group: integers modulo n. The multiplicative group of invertible classes modulo n. Finite fields. Construction of finite fields by extension of a known finite field.

Subgroups: equivalent definitions. Examples. 

Cyclic groups and subgroups. Order of an element.

Group homomorphism, isomorphism, automorphism. Internal automorphism.

Dihedral groups.

Symmetric groups: isomorphism with permutation matrices. Sign of a permutation. Decomposition into disjoint cycles. The alternating subgroup.

Subgroup cosets. Index of a subgroup. . Lagrange theorem. Normal subgroups

Quotient groups. Kernel and image of a homomorphism

Homomorphism theorems.

Group actions. Orbits and stabilizers. Sylow theorems. The free group. Presentation of a group by generators and relations.

Readings/Bibliography

G. M. Piacentini Cattaneo: ALGEBRA, un approccio algoritmico,

Zanichelli, 1996

I.N. Herstein: Algebra. Editori riuniti, 2010.

M.Artin: Algebra. Bollati Boringhieri 1997.

S. Lang: Undegraduate Algebra, Springer-Verlag, 1987

Teaching methods

Traditional lectures in classroom with the possibility for students in need to follow lectures online. At least two hours per week are dedicated to exercises.

Assessment methods

The goal of the final examination is to evaluate the achievement of the following objectives: deep knowledge of the algebra tools presented during the course; ability of using these tools to solve a problem in algebra.
The final examination consists of a written exam and an oral exam. To attend any of the exams the student has to register on the web-site AlmaEsami.
The written exam consists of exercises and problems in abstract algebra. No theoretical question will be asked in this part of the examination. During the written exam books and notes are not allowed. The evaluation consists in a numerical note with a maximum of 30 points. At least 15 points are requested to be admitted to the oral exam. Results of the examination are posted on the web-site AlmaEsami.
The goals of the oral exam is to evaluate the theoretical knowledge of the subject. Before the oral exam there will be a discussion on the board about the written exam. The final note will consider of both written and oral exam and its registration will be performed after the oral exam.
The students have six opportunities to attend the exam during the academic year and their exact dates will be available in large advance.

Office hours

See the website of Fabrizio Caselli

See the website of Luca Moci