SCC0050 - ALGEBRA 1

courses

ID:

SCC0050

Duration (hours):

86

CFU:

SSD:

ALGEBRA

Year:

2025

Date/time interval

Primo Semestre (22/09/2025 - 16/01/2026)

Course Objectives

TEACHING OBJECTIVES The aim of the course is to provide the students with a rigorous settlement of well-known arithmetic results and to introduce them to the study of abstract algebraic structures EXPECTED LEARNING OUTCOMES At the end of the course the student will be able to: Use the main algorithms in natural numbers arithmetic and justify them. Use an adequate mathematical language and expose proofs in a coherent and conscious way Establish whether a given operation satisfy certain properties Establish whether a given algebraic structure satisfy certain axioms

Course Prerequisites

Standard high school mathematics.

Teaching Methods

Frontal lectures. Attending lectures is not mandatory but strongly recommended. During lectures, the teacher uses either a board or a tablet to write down what is being explained. Unless there are technical issues, the content written on the tablet is later made available to students through the course website and/or the e-learning platform. Every topic is explained together with exercises useful to understand and apply exposed results. Sometimes the solution is given immediately, sometimes in a subsequent lecture to stimulate students to autonomous work. Especially at the end of the course, lectures of summarizing exercises are scheduled, to help students to choose the more appropriate method to solve an exercise and to establish connections between different topics.

Assessment Methods

Written and oral exam.
The written examination lasts 2 hours and 30 minutes and tipically consists of 4 or 5 exercises divided in subquestions. It is required the ability to establish properties of the algebraic structures and to justify all the statement given.
The written examination is graded on a scale from 0 to 30. Students need a grade of 14/30 or more to be admitted to the oral examination.

The oral examination usually begins with the discussion of the written test. The student will then be required to present some of the results seen in class. It will be object of evaluation the ability to present a proof in a complete and rigorous.

A written mid-term examination is reserved to first year students: this proof will be graded from 0 to 5 to be added to the grade of the ordinary written examination.
Passing examination and the final grading depend both on oral and written tests. More detail on course web page.

SETS. Correspondences. Mappings. Composition of mappings. Associativity of composition. Injective, surjective e bijective mappings. Inverse mapping of a bijective mapping. Relations in a set. Equivalence relations. Partial and total order relations. MATRICES. Matrix operation. Transpose of a matrix. Diagonal and triangular matrices. INTEGERS. Operations and their properties. Ordering. Principle of induction. Division. Highest common factor. Bezout identity. Prime numbers characterization. Euclidean algorithm. Prime factorization. Existence of infinite primes. Congruences modulo an integer. Residue classes and operations. The ring of the residue class. Cancellation law and invertible elements modulo n. Linear congruence equations. Chinese remainder theorem. Euler's function. GROUPS. Binary operations. Monoids. Groups. Commutative monoids and groups. Plane transformations. Klein's group. Dihedral group. Cancellation laws. Multiplication table. Subgroups. Criterions for subgroups. Intersection of subgroups. Subgroup generated by a subset. Explicit description of the element of the subgroup generated by a subset. Centralizer of an element of a group. Center of a group. Symmetric group. Cycles. Disjoint cycles. Transposition. Parity of permutations. Alternating subgroup. Powers in groups and monoids. Order of an element. Cyclic groups. Subgroups of a cyclic group and their reciprocal position. Number of elements of a given order in a cyclic group. Right and left cosets of a subgroup. Index of a subgroup. Lagrange's theorem. Product of subgroups. Group and monoid homomorphism. Kernel and image of a homomorphism. Direct and inverse images of subgroups. Homomorphisms from cyclic groups. Endomorphisms of cyclic groups. Action of a group on a set. Trivial action. Orbits and stabilizers. Orbit equation. Conjugation in a group and conjugacy classes. Center of a finite p-group. Normal subgroups. Conjugacy classes in the symmetric and alternating groups. Congruences in monoids and groups. Quotient monoids and groups. Congruences and normal subgroups. Quotient over the center of a group. Isomorphism theorems. Product of normal subgroups. Internal and external direct product. Direct product of cyclic groups. Internal and external semidirect product. Reversal of Lagrange's theorem for abelian groups and p-group. Sylow theorem with application to classification problems. Classification of groups of order a square of a prime.

Course Language

Italian

More information

For simple and short questions, ask the teacher immediately before or after the class. Email teacher for longer questions or for appointments (valerio.monti@uninsubria.it). For further detail go to the web page of the course (algebrainsubria.altervista.org).