October 2, 2017: We have briefly overviewed the history of the development of Algebra. We defined the cartesian product of sets and the n^{th} cartesian power of a set for a non-negative integer n. We defined and studied some common kinds of binary relations as equivalences, partial orders and preorders. In particular, we described the connection between equivalences and partitions of a given set.
October 9, 2017: We defined an operaton on a set as a map from its cartesian power to itself and an algebra as a set equipped with operations. We studied algebras with a single binary operation. In particular, we defined grupoids, semigroups, monoids, loops, and groups. We studied properties of unit elements and inverses.
October 16, 2017: We defined the symmetric group S_{n} of all permutations of an n-element set. We proved that every permutation decomposes as a product of indpendent cycles. We defined the signum of a permutation as (-1)^{ n - #cycles } and proved that every permutation is a product of transpositions.
October 23, 2017: We stated that the signum of a permutation equals (-1)^{ #transpositions } in any decomposition of the permutation into the product of transpositions. We concluded that products of even permutations are even, and that even permutations form a subgroup, called the alternating group A_{n}. We studied the 15 puzzle and we proved that positions corresponding to odd permutations are unsolvable.
October 30, 2017: We defined left an right cosets of a subgroup. We shoved that they correspond to the blocks of a congruence relation modulo the subgroup. We proved that all left (right) cosets of a subgroup H have the same size. The size is equal to the size of H (which, indeed, is one of the cosets). We defined an index [G:H] of a subgroup H in a group G as the number of all left (equaly the number of all right) cosets of H. We finished the lecture with the Lagrange theorem that |G| = [G:H]|H|.
November 6, 2017: A subgroup N of a group G is normal if left and right cosete of N coincide. We proved that for a normal subgroup N of a group G we can define a factor-group G/N whose elements are cosets of N. We defined the equivalence ~ of conjugacy on G by g~h if g = f.h.f^{ -1} for some f ϵ G. We proved that congruences are conjugated if and only if they are of the same type (= cycle structure). We proved that the alternating group of permutations A_{n} is simple (has no non-trivial normal subgroup) for all n>4.
November 13, 2017: We defined a group homomorphism, group embedding, and group isomorphism. We proved that a group homomorphism is an isomorphism if and only if it has an inverse. We defined the kernel of a group homomorphism and we proved that the kernels are exactly normal subgroups. We stated and proved The first homomorphism theorem.
November 20, 2017: We proved three isomorphism theorems for groups.
November 27, 2017: We defined and studied congruences modulo a positive integer n. We found all subgroups of the group Z of all integers and we understood how the congruences modulo n correspond to congruences modulo the subgrous. We described all cyclic groups up to isomorphism. We proved that subgroups and factorgroups of cyclic groups are cyclic. We defined the Euler φ-function that assigns to a positive integer n the number of generators of the n-element cyclic group and we showed how to compute it.
December 4, 2017: We defined an action of a group G on a set X, isotropy subgroups, and G-orbits. We proved Burnside's theorem and gave examples of its applications due to Pólya. We proved the Class formula and used it to prove Cauchy's theorem.