### Materials

Lecture 1 (07.01): Symmetry group of an equilateral triangle. Definition of a group, Abelian group, semigroup and monoid. Examples of groups. Uniqueness of identity and inverse elements. Cayley table. Definition and examples of subgroups. (Metsänkylä-Näätänen: III.1-III.2 until Esimerkki 4)

Lecture 2 (09.01): Characterization when a subset is a subgroup (three or two conditions). Characterization when a finite subset is a subgroup. Definition of the center of a group. The center of a group is a subgroup. The intersection of subgroups is a subgroup. Definition of the subgroup generated by a set together with examples. Characterization of the subgroup generated by one element. Definition of finite and infinite order. Definition of the order of an element. Further characterizations of the subgroup generated by one element for elements of finite and infinite order. (Metsänkylä-Näätänen: III.2-III.3)

Lecture 3 (14.01): Definition and examples of a cyclic group. If a group is cyclic, then it is abelian. Definitions of a relation, an equivalence relation, an equivalence class and a partition. Correspondence between partitions and equivalence relations of a set. Right and left congruence relation on a group determined by a subgroup. These relations are equivalence relations. Left and right cosets. Index of a subgroup. Lagrange's Theorem. (Metsänkylä-Näätänen: II.6, III.5 until Lause 12)

Lecture 4 (16.01): Left and right multiplication are bijections on a group. The cardinality of each coset is equal to the cardinality of the subgroup. Proof of the Lagrange's theorem. The order of an element of a group divides the order of the group. Each element raised to the power of |G| gives the identity. Every group of prime order is cyclic. Congruence modulo n is an equivalence relation; Zn as the set of its equivalence classes; addition and multiplication on Zn; the group of units of Zn. Fermat's little theorem. Classification of groups of order at most 5. Definition of a group homomorphism, endomorphism, isomorphism and automorphism. Examples of homomorphisms. Properties of group homomorphisms. (Metsänkylä-Näätänen: III.5, III.4 until Lause 8)

Lecture 5 (21.01): Isomorphisms: inverse of an isomorphism is an isomorphism; Aut(G) with composition is a group; examples. Kernel and image of a homomorphism. Kernel and image are subgroups of G and H respectively. Characterizations of injective and surjective homomorphisms using kernel and image. Normal subgroup; examples: {e} and G, every subgroup of an abelian group, the subgroup of rotations of D3. A characterization of normal subgroups. Kernel of a homomorphism is a normal subgroup. Simple groups. The set G/H and a binary operation of G/H (H has to be normal). (Metsänkylä-Näätänen: III.4, IV.1 until Lause 2)

Lecture 6 (23.01): Quotient group. First, second and third isomorphism theorem for groups. (Metsänkylä-Näätänen: IV.1-IV.2)

Lecture 7 (28.01): Cyclic groups. First we discussed group homomorphisms from cyclic groups and as a corollary showed that two cyclic groups are isomorphic if and only if their cardinalities are equal. We also characterized all subgroups of infinite cyclic groups and as an example considered the subgroups of (Z,+). Then we showed that if a is an order n element of an arbitrary group G, then the order of a^m is n/gcd(n,m). We stated but didn't prove yet the characterization of all subgroups of finite cyclic groups. If you missed today's lecture, please read Metsänkylä-Näätänen Chapter IV.3 or Reis-Rankin Chapter 5.5.

Lecture 8 (30.01): We proved the characterization of all subgroups of finite cyclic groups. As a corollary, every subgroup of a cyclic group is cyclic. We defined a permutation and showed that all permutations of n elements form a group under composition. This group is called a symmetric group (denoted S_n) and its subgroups are called permutation groups. We defined an r-cycle and showed that every permutation can be written as a composition of disjoint cycles. We defined the type of a permutation. We showed that if a permutation has type (r1,r2,...,rm), then the order of the permutation is lcm (r1,r2,...,rm). We defined a transposition and remarked that transpositions generate the symmetric group S_n. We defined a sign of a permutation and showed that sign is multiplicative. Hence sign is a homomorphism from S_n to {1,-1}. The kernel of sign is called the alternating group. We stated and proved the Cayley's theorem. (Metsänkylä-Näätänen Chapter IV.4)

Lecture 9 (04.02): We defined a ring and a commutative ring. We considered examples of rings. We defined a unit in a ring and considered examples. We studied basic properties of a ring. We defined a subring and discussed subring criterium. (Metsänkylä-Näätänen V.1-V.3 until Lause 4).

Lecture 10 (06.02): We defined an ideal. We gave the ideal criterion. We stated that any intersection of ideals is an ideal and a finite sum of ideals is an ideal. We defined the ideal generated by a set S, what it means to be finitely generated and a principal ideal. We gave a characterization of an ideal that is generated by finitely many elements in commutative rings. We defined principal ideal rings. We defined a ring homomorphism. (Metsänkylä-Näätänen Chapter V.3, V.5 until Esimerkki 4).

Lecture 11 (11.02): We discussed basic properties of ring homomorphisms. We defined quotient rings and proved the (first) isomorphism theorem for rings. We defined a zero-divisor, an integral domain and characteristic. We discussed the cancellation law in integral domains and showed that in an integral domain all non-zero elements have the same order. (Metsänkylä-Näätänen Chapter V.4-V.6).

Lecture 12 (13.02). We showed that the characteristic of an integral domain is 0 or a prime number. We defined a field. We showed that every field is an integral domain and every finite integral domain is a field. We showed that the only ideals of a field F are {0} and F and that every field homomorphism is injective. We defined a subfield and gave the subfield criterion. (Metsänkylä-Näätänen Chapter VI.1-VI.2). We also discussed the RSA algorithm as an application of the group theorem (not on the exam or in the materials). We did not cover all of Metsänkylä-Näätänen Chapter VI.2 and you are encouraged to read it on your own, although it won't be on the exam.