# Algebra I

### Abstract

We will introduce algebraic structures as sets with operations of various arities subject to certain axioms. We study groups as the first example. We prove some standard results as the Lagrange theorem or the simplicity of the group A5. We investigate cyclic groups and we apply our results to find simple proofs of some standard number theoretic theorems. We study operations of groups on sets and we introduce Polya's theory of counting. Then we pass to commutative rings and divisibility. We define and study PIDs and UFDs. We will apply the results to solve some non-trivial Diophantine equations. We will conclude the course with the introduction to the theory of fields.

The exam will be oral. It will consists of three questions; one general asking to overview certain topic, for example, symmetric groups or unique factorization domains, one more specific, for example, to formulate and prove the Lagrange theorem for groups, and one testing the understanding, for example, to find a non-trivial normal subgroup of the group A4. Students will get time to prepare the answers.

### Lectures

#### Relations and operations on sets

1. October 2, 2017: We have briefly overviewed the history of the development of Algebra. We defined the cartesian product of sets and the nth 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.

2. 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.

3. #### Groups of permutations

4. October 16, 2017: We defined the symmetric group Sn 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.

5. 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 An. We studied the 15 puzzle and we proved that positions corresponding to odd permutations are unsolvable.

6. #### Basics of group theory

7. 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|.

8. 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 An is simple (has no non-trivial normal subgroup) for all n>4.

9. 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.

10. November 20, 2017: We proved three isomorphism theorems for groups.

11. #### Cyclic groups

12. 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.

13. #### Groups acting on sets

14. 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.

15. #### Rings, ideals and divisibility

16. December 11, 2017: We defined rings and ideals, we proved that ideals correspond to kernels of ring homomorphisms. We studied divisibility in commutative cancellative monoids. We proved that if greates common divisors exist, then every irreducible element of the monoid is prime.

17. December 18, 2017: We studied connections between divisibility and order of principal ideals in a ring. We defined principal ideal domains and we proved that in principal ideal domains greatest common divisors exist. We defined Euklidean domains and we proved that they are principal ideal domains. Finally we showed that the ring Z[i] of Gaussian integers is an Euklidean domain.

18. January 8, 2018 : We studied Gaussian monoids and defined unique factorization domains as integral domains with monoid of non-zero elements Gaussian. We proved that every principal ideal domain is a unique factorization domain. We showed that in the domain Z [(-3)½] contains irreducible elements that are not prime. Consequently it is not a unique factorization domain. Finally, we characterized primes in the ring Z [i] of Gaussian integers and solved some non-trivial problems using decompositions in Z [i].

### Bibliography

1. Lang, S., Algebra (Rev. 3rd ed.), Springer-Verlag, 2002. (Chapters I,II,V.)

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