Lecture:
Tuesday 9:00 - 10:30 S11
Tutorial:
Tuesday 10:40 - 12:10 S11
Abstract. We introduce algebras to be sets with operations of various arities subject to axioms. We clasify algebras with a single binary operation. Then we study groups. We prove basic results, in particular the Lagrange theorem and the simplicity of the group A5. We investigate cyclic groups. Applying the results to find simple proofs of some standard number theoretic theorems. We study acting of groups on sets. As an application we introduce Polya's theory of counting. Then we turn our interest to commutative rings. We introduce principal ideal domains and unique factorization domains study. We take a closer look at the domain of gaussian integers. We apply the results solving Diophantine equations. We conclude the course with the introduction to the theory of fields.
Credit and exam. The credit will be given for solving at least 50% of the homeworks which means to get at least 10 points. The exam will be oral consisting of three questions. Here is an example.
October 2, 2018. Lecture: We defined the cartesian products and the nth Cartesian powers of sets. An n-ary relation on a set is a subset of its nth Cartesian power. We studied the most common properties of binary relations, namely reflexivity, transitivity and various kinds of symmetries and anti-symmetries. Via these properties we defined equivalences, partial orders and preorders. In particular, we described the connection between equivalences and partitions of a given set. Tutorial: We studied properties of relation by means of the operations of the composition and the transpose.
October 9, 2018. Lecture: An n-ary operation on a set as a map from its nth Cartesian power to the set. An algebra is a set equipped with operations (possibly of various arities) subject to some relations. We focused on algebras with a single binary operation. In particular, we defined grupoids, semigroups, monoids, loops, and groups. Tutorial: We recalled some basic properties of permutaions. In particular, we proved that each permutation of a finite set decomposes uniquelly as the product of independent cycles (up to their order). The first homework is Exercise 2.4.
October 16, 2018. Lecture: We defined notions of a sub-universe and a sub-algebra of an algebra. We discussed these notions in the particular case of groups. Then we went back to study of permutations. We defined the signum of a permutation by means of the number of its cycles and we proved that the signum can be computed from any decomposition of the permutation into a product of transpositions. Tutorial: We introduced the 15 puzzle and we proved that positions with the empty space in the right bottom square corresponding to odd permutations are unsolvable. The second homework is Exercise 3.6.
October 23, 2018. Lecture: We defined right and left cosets of a subgroup of a group. We proved that all of them have the size equal to the size of the subgroup. We call the number of the left cosets the index of the subgroup. We proved the Lagrange theorem that the size of a group equals the size of a subgroup times the index of the subgroup. We defined a normal subgroup of a group and we showed various equivalent characterizations of normal subgroups. Tutorial: We studied relations between left and right cosets of a subgroup and properties of normal subgroups. The third homework is Exercise 4.1.
October 30, 2018. Lecture: We studied the conjugacy relation on a group. We proved that the conjugacy classes form a partition of the group. We showed that a subgroup is normal if and only if it is a union of conjugacy classes. Further we proved that permutations in a symmetric group are conjugated if and only if they have the same type. Finally we studied sets of generators of a group. Tutorial: We studied conjugacy classes of symmetric and alternating groups and groups of symmetries of geometric objects. We described some sets of generators of symmetric and alternating groups. We applied this and showed that all possitions of the 15 puzzle corresponding to even permutations are solvable. The fourth homework is Exercise 5.5.
November 13, 2018. Lecture: We defined and studied the notions of a group homomorphism and the kernel of a group homomorphism. We proved that the kernels correspond to normal subgroups. Tutorial: We examined examples of group homomorphisms. The examples were based on a geometric intuition. We also classified normal subgroups of the symmetric group S4. The fifth homework is Exercise 6.2.
November 20, 2018. Lecture: We proved the homomorphism theorem and the three isomorphism theorems for groups. Tutorial: We applied the homomorphism and the isomorphism theorems to prove some relations among cyclic groups. The sixth homework is Exercise 7.1.
November 27, 2018. Lecture: We characterized all cyclic groups up to isomorphism and proved that a subgroup and a factor group of a cyclic group is cyclic. We proved that a cyclic group of a finite order n contains a unique subgroup of order m for every divisor m of n. We studied orders of products of commuting elements of a group. We proved that a finite subgroup of a multiplicative group of all non-zero elements of a field is cyclic. Then we studied the sets of generators of cyclic groups. We computed the Euler function and proved the Euler's, the (small) Fermat's, and the Wilson's theorems. Tutorial: We studied properties of congruences modulo positive integers, orders of elements of groups, and properties of the Euler function. The seventh homework is Exercise 8.6.
December 4, 2018. Lecture: We studied groups acting on sets. We called sets equipped with an action of a group G-sets. We defined an orbit and a stabilizer of an element of a G-sets. We proved the Burnside's lemma and the class formula and we showed some applications of these results. In particular, we introduced some ideas of the Pólya's theory of counting. Tutorial: We studied coloring problems solvable by the Pólya's theory of counting and, in a sequence of steps, we proved the simplicity of the group A5. The eighth homework is Exercise 9.5.
December 11, 2018. Lecture: We defined rings and showed some examples. We introduced the notion of an ideal of a ring. We proved that ideals correspond to kernels of ring homomorphisms. Finally we studied the divisibility in commutative cancellative monoids. Tutorial: We studied various constructions of rings. In particular, we investigated polynomial and matrix rings. The ninth homework is Exercise 10.1.
December 18, 2018. Lecture: We studied the relationship between divisibility in commutative domains and the ordering of their principal ideals. We defined the notion of a principal ideal domain and we proved that greatest common divisors and least common multiples exist in principal ideal domains. We introduced Euclidean domains and we proved that Euclidean domains are principal ideal domains. Finally we studied the ring Z[i] of Gaussian integers and we proved that Z[i] is an Euclidean domain. Tutorial: We discussed the Axiom of Choice and proved the equivalence of the Axiom of Choice and the Zorn`s lemma. The tenth homework is Exercise 11.1.
January 8, 2019. Lecture: We defined Gaussian monoids as commutative cancellative monoids with unique decomposition into products of irreducibles and we characterized them via the existence of greatest common divisors and decreasing chain conditions of the relation of divisibility. We applied this characterization to prove that principal ideal domains are unique factorization domains. Tutorial: We proved that the ring Z[i] of Gaussian integers is a unique factorization domain. We applied this fact when solving some Diophantine equations.
