Subject: ALGEBRA A (A.A. 2020/2021)
Unit Algebra A
Basic Mathematics (lesson)
Objectives of course is to furnish the basic tools for the study of algebra, to study some basic algebraic structures and some of their applications. Objectives of course is to let the students get familiar with abstract mathematical arguments.
Set theory:symbolism, terminology, sets computations. Numerical sets: Natural numbers, Fractional numbers, Real numbers, Complexes numbers. Algebraic computation in the real set numbers.
GROUPS: Definitions and basic facts. Examples. Subgroups. Abelian groups. Finite groups. Symmetric group and alternating group. Rotation group and dihedral group. Cyclic groups. Cosets of a subgroup. Lagrange's theorem, Sylow's theorem, Cauchy's theorem. Normal subgroups. Simple groups. Alternating groups for n>4 are simple. Quotient group. Homomorphisms, isomorphisms. Foundamental theorems on homomorphisms. Centre of a group. Automorphisms and inner automorphisms. Cayley's theorem. Characteristic subgroups. Commutators and derived subgroups. Solvable groups. Chain of subgroups. Non-solvability of symmetric group for n>4. Every finite p-group (p prime) is solvable. Every group of order p^2 (p prime) is abelian. Caracterization of simple solvable groups. Direct product of groups. LATTICES: Order relationes. Lattice definition and examples. Sublattice. Modular lattices. Distributive lattice. Isomorphisms and antiisomorphisms. Boolean algebras. Chain of elements of a lattice. Maximal chains and Jordan-Dedekind condition.
Frontal lectures which include theory and exercizes.
End of course oral assessment.
Total compliance with the Dublin descriptors.
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