# ALGEBRA 1

Degree course:
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree:
2017/2018
Year:
1
Academic year in which the course will be held:
2017/2018
Course type:
Basic compulsory subjects
Credits:
8
Period:
First Semester
Standard lectures hours:
48
Detail of lecture’s hours:
Lesson (48 hours)
Requirements:

Standard high school mathematics.

Final Examination:
Orale

A partial written examination is scheduled in november. The student will obtain from 0 to 4 points to be added to the grading of the first 2 exam sessions.

The written examination lasts 2 hours and 30 minutes and tipically consists of 4 or 5 exercises divided in subquestions.
The written examination is graded on a scale from 1 to 30. Students need a grade of 14/30 or more to be admitted to the oral examination.
Passing examination and the final grading depend both on oral and written tests.

Assessment:
Voto Finale

Knowledge of basic algebraic structures and abstract properties of operations.

Sets 8 hours
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.

Integers. 16 hours
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. 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 40 hours
Binary operations. Monoids. Groups. Commutative monoids and groups. Residue classes. Plane transformations. Klein's group. Dihedral group. Cancellation law. Multiplication table.
Subgroups. Criterions. Intersection of subgroups. Subgroup generated by a subset.
Action of a group on a set. Trivial action. Transitive actions. Orbits and stabilizers. Orbit equation. Conjugation in a group: conjugacy classes. Centralizer of an element. Center of a finite p-group. Classification of groups of order a square of a prime. Right and left cosets of a subgroup. Index of a subgroup. Lagrange's theorem.
Symmetric group. Cycles. Disjoint cycles. Conjugacy classes in the symmetric group. Transposition. Parity of permutations. Alternating subgroup.
Powers in groups and monoids. Cyclic groups. Order of an element. Subgroups of a cyclic group and their reciprocal position. Number of elements of a given order in a cyclic group.
Normal subgroups. Quotient group. Quotient over the center of a group.
Group and monoid homomorphism. Kernel and image of a homomorphism. Direct and inverse images of subgroups and normal subgroups. Isomorphism theorems. Homomorphisms from cyclic groups. Endomorphisms of cyclic groups.
Product of subgroups. Product of normal subgroups. Internal and external direct product. Direct product of cyclic groups.

P.M. Cohn, Classic Algebra, Wiley.

Exercises, exam problems, and lecture notes available on the class web page.

Frontal lectures and guided exercises

For further detail go to the web page of the course.