Office hours: By appointment
Jan 9th: Some broad ideas such as symmetries of objects, operations, and equations and their solutions were discussed. Then several examples of sets with binary operations were given and discussed. Among the examples were addition and multiplication of numbers and matrices, permutations, composition of functions, unit circle, and roots of unity. Also associative binary operations were discussed.
Jan 13th: Semigroups and groups were defined, sevral examples were given, and some of their basic properties were discussed.
Jan 16th: The group generated by a subset of a group was introduced. Then cyclic groups were introduced and it was shown that any subgroup of a cyclic group is cyclic; also some related examples were discussed.
Jan 20th: The center of a group and the centralizer of an element in a group were itroduced and it was shown that they form subgroups. Then Quaternion group was introduced and discussed. Then the structure of cyclic groups was analysed precisely. Also the order of an arbitrary element in a finite cyclic group was computed.
Jan 23rdh: The cosets of a subgroup H in a group G were indtroduced and the Lagrange Theorem was proved, namely that it was shown that the order of any subgroup of a finite group divides the order of the group. Also some related notions such as the index of a subgroup in a group were introduced and their basic properties were discussed.
Jan 27th: It was observed in an example that the product of two suggroups of a group is not necessarily a subgroup, and a necessary and and sufficient condition for this to hold was proved. Then a formula was proved which says precisely how many elements exsit in the product set of two subgroups of a group. Then few examples related to this and the cosests of subgroups were discussed.
Jan 30th: It was proved that the intersection of finitely many subgroups of a group that have finite index in a group has finite index, which is Poincare's Theorem. Then normal subgroups were introduced and sevral examples were given. It was shown that the cosets of a normal subgroup admit a natural group structre.
Feb 3rd: First some remarks about normal subgroups were given. Then a theorem was proved, which states that if the index of a subgroup of a finite group is the smallest prime that divides the order of the group, then the subgroup is normal. Then the notion of normalizer of a set in a group was introduced and discussed. Also the derived group of a group was introduces and some of its basic properties were mentioned.
Feb 6th: First it was proved that if the quotient group G/Z(G) of a group by its center is cyclic, then the group G is abelian. Then, after discussing some basic properties of group homomorphisms, the first and second isomorphism theorems were stated and proved. At the end the third isomorphism theorem was stated.
Feb 10th: The third Isomorphism Theorem was proved. Then the group of automorphism Aut(G )of a group was interoduced and it was shown that inner automorphisms Inn(G) form a normal subgroup of Aut(G). It was also shown that G/Z(G) is isomorphic to Inn(G). At the end a correspondence theorem was proved, which characterizes all subgroups and normal subgroups of the quotient of a group by a normal subgroup.
Feb 13th: The conjugacy relation in groups was introduced and it was shown that the number of elements in the class of an element in a group is equal to the index of its centralizer in the group. Then the class equation for finite groups was derived. At the end it was shown that if the order of a group G is a power of a prime number, then Z(G) is nontrivial, any nontrivial normal subgroup of G intersects with Z(G) nontrivially, and any proper subgroup of G is a proper subgroup of its normalizer in G.
Feb 24th: It was proved that any group with a p^2 elements, where p is a prime number, is abelian. Then the notion of conjugate subgroups was introduces and it was shown the number conjugates subgroups of a subgroup H of a group G is equal to the index of the normalizer of H in G. Then maximal and maximal normal subgroups were introduced and a few examples were given. At the end it was shown that if H and K are two distinct maximal normal subgroups of a group G, then the intersection of H and K is maximal normal in both H and K.
Feb 27th: The symmetric group on a set and permutation groups were introduced and a few examples of permutations, cycles, etc and the center of S_n were discussed. Then the Cayley theorem was proved, which states that any group is isomorphic to a permutation group. Then the notion of the core of a subgroup was introduced and it was proved that if H is a subgroup of G whose index is n, then there exists a group homomorphism from G to S_n whose kernel is the core of H.
March 3rd: The cycle decomposition for permutations was proved, and as a corollary it was shown that any permutation is a product of transpositions. Then the cycle structure of permutations and its invarinace under conjugacy was discussed, and it was shown that the number of partitions of a positive integer is equal to the number of conjugacy classes in S_n.
March 6th: It was proved in two different ways that if a permutation in S_n is a product of r transpositions and a product of s transpositions, then r and s are both even or both odd. Then the alternating group of degree n, A_n, which is the set of even permutations in S_n, was introduced and it was shown that it is normal in S_n. Then few examples were discussed, in particular the alternating groups of degree 3 and 4 were explained in detail. We observed that A_4 has only one normal subgroup, which is of order 4, and concluded that the opposite of the Lagrange theorem is not true as 6 divides the order of A_4, but A_5 does not have any subgroup of order 6. At the end it was shown that cycyles of length 3 generate A_n and that the derived group of S_n is A_n.
March 10th: First a short discussion about simplicity of A_n, n greater than 4, was given and it was mentioned that in this case S_n is not a solvable group. Then the notion of group action was introduced and several examples were given. Then the Burnside theorem was proved, which shows that if a finite group G acts on a finite set, then the number of orbits is equal to the average over G of the size of fixed points of g in G. At the end we started the subject of the structural theorem of groups.
March 13th: The fundamental theorem of finitely generated abelian groups was proved. That is, it was shown that any such group is the direct sum of finitely many cyclic subgroups. It was shown that the finite cyclic groups in the direct sum can be ordered in a way that the order of the first one devides the order of the second one, which devides the order of the third one and so on so forth.
March 17th: It was shown that the list of consecutively dividing integers which appear in the decomposition of a finite abelian group into the direct sum of cyclic groups is an invariant of the group. Then another uniqueness theorem related to the structure of finite abelian groups was proved, which states that if p_i are the distinct primes in the prime decomposition of the order of an abelian group, the group is the direct sum of the subgroups H_i defined to be the set of elements whose orders are powers of p_i. This theorem allows one to find all non-isomorphic ablelian groups of any arbitrary finite order.
March 20th: Cauchy's theorem was proved, which states that if a prime p divides the order of a finite group, then there is an element of order p in the group. Then sylow subgroups and p-groups, p a prime, were defined and the first sylow theorem was stated, which is a generalization of Cauchy's theorem.
March 24th: Sylow's first theorem was proved and a few of its colollaries were discussed. Then a second proof for Cauchy's theorem was given, which uses group actions.
March 24th: Sylow's first theorem was proved and a few of its colollaries were discussed. Then a second proof for Cauchy's theorem was given, which uses group actions.
March 31st: The second and third Sylow theorems were proved.
April 3rd: Several examples for applications of Sylow theorems were given.
April 7th: A couple of examples related to applications of Sylow theorems were given. At the end, the Dihedral groups were indtroduced and their presentation by generators and relations was derived.