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

SETS

Cartesian products, set of subsets. Relations, equivalence relations; partial and total order relations. Hasse diagrams. Divisibility and congruence between integers. Equivalence classes, quotient sets. Set partitions and their equivalence with equivalence relations. Maps. Injective, surjective and bijective maps.Elementary theory of cardinality. Elementary  enumerative combinatorics.

NUMBERS

Natural numbers, construction of integer and rational numbres. Prime numbers and irreducible numbers. Division with remainder in Z. The GCD in Z. EUclid's algorithm. Bézout's identity. Irreducibles are prime. Well-order principle. Induction principle. Divisibility criterions, other basis expressions. Sum and product in Z/n. Construction os integers and rational numbers as equivalence classes. Fundamental theorem od arithmetic.

GROUPS

Definition. Commutative groups. Bijections of a set, isomorphisms of a vector space. Homomorphisms and isomorphisms. Subgroups. Order of an element. Direct product of groups. Cancellation law. Standard notation for abstrac groups. Powers of an element. Classification of subgroups of Z. Classification of subgroups of Z/n. Intersections and union of subgroups. Subgroup generated by a subset and description of its elements. Cyclic groups, their classification and their generators.

PERMUTATION GROUPS

Dihedral groups: rotations and reflections. Generators. Relations. Th symmetric groups. One line and two line notations. Composition, Action on {1,2,...,n}. Orbits. Cycles. A permutation is the product of its cycles. Transpositions. Even and odd permutations. Inversions of a permutation. The sign of a permutation. The alternating groups. Conjugation in S_n. The permutations are conjugate if and only if they have the same cycle structure.

COSETS and QUOTIENTS

Left and right cosets. The index of a subgroup. Bijection between left and right cosets and between two cosets. Lagrange theorem. Kernet and image of a homomorhism. Cayley's theorem. Normal subgroups. Center. Characterization of normal subgroups. Quotinet groups and projections. Compatible relations: correspondence between compatible relations and normal subgroups. Homomorphism's theorem.

GROUP ACTIONS

Internal and external direct products. Internal semidirect products. External semidirect products. Classification of normal subgroups of a quotient. Groups action on a set. Conjugation on the group and on its subgroups. Action by left multiplication, onthe group, on the cosets of a subgroup, on the set of subsets of given cardinality, Orbits and stabilizers. Formula of orbits. Formula of classes and its consequences. Cauchy's theorem. Sylow's theorem: 3 proofs.

 

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