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## Math 103A Fall 2019 Modern Algebra I ### Lecture Notes

• Lecture 1  (Friday 09/27/19) Group axioms, examples of additive/multiplicative groups, matrices.
• Lecture 2  (Monday 09/30/19) Integers, division with remainder, GCD, Euclid's algorithm.
• Lecture 3  (Wednesday 10/02/19) Congruences modulo N, residue classes, partitions of Z.
• Lecture 4  (Friday 10/04/19) Modular arithmetic, multiplicative inverses modulo N, Euler's totient.
• Lecture 5  (Monday 10/07/19) Composition tables, examples, cancellation laws, multiplication-by-a.
• Lecture 6  (Wednesday 10/09/19). No duplicates in rows and columns, permutations of G, Cayley's theorem (v.1).
• Lecture 7  (Friday 10/11/19). Chinese remainder theorem, its group reformulation, computing Euler's phi.
• Lecture 8  (Monday 10/14/19). Powers, order of an element a, the subgroup generated by a, cyclic groups.
• Lecture 9  (Wednesday 10/16/19). Cyclic groups, generators, primitive roots mod p, examples Z and Z_N.
• Lecture 10  (Friday 10/18/19). Orders of powers formula, number of generators is Euler's phi, Nth roots of unity U_N.
• Lecture 11  (Wednesday 10/23/19). Identifying U_N with Z_N. Classification of cyclic groups up to isomorphism, subgroups.
• Lecture 12  (Friday 10/25/19). Subgroups of cyclic groups are cyclic, the divisor correspondence.
• Lecture 13  (Monday 10/28/19). Example: Subgroups of Z_45, the sum formula for phi, primitive roots mod p continued.
• Lecture 14  (Wednesday 10/30/19). Dihedral groups, review of rotations and reflections of the plane, SO(2).
• Lecture 15  (Friday 11/01/19). Dihedral groups continued, SO(3), rotational symmetries of the cube and tetrahedron.
• Lecture 16  (Monday 11/04/19). The group SO(3) and rotations of the Platonic solids continued.
• Lecture 17  (Wednesday 11/06/19). Permutations, the symmetric group S_n, identifying S_3 with D_3, cycles, transpositions.
• Lecture 18  (Friday 11/08/19). Decomposition into disjoint cycles, order is LCM of lengths, sign of a permutation, A_n.
• Lecture 19  (Friday 11/19/19). The sign homomorphism, permuting variables of polynomials, signs via the number of crossings.
• Lecture 20  (Monday 11/18/19). Left and right cosets, the index [G:H], Lagrange's theorem.
• Lecture 21  (Wednesday 11/20/19). Examples of cosets, corollaries of Lagrange, Euler's congruence and Fermat's little theorem.
• Lecture 22  (Friday 11/22/19). Normal subgroups, group structure on G/H, examples.
• Lecture 23  (Monday 11/25/19). More on normal subgroups, V_4 in A_4, no H in A_n of index two, homomorphisms.
• Lecture 24  (Wednesday 11/27/19). Image and kernel, ker(f) is normal, examples.
• Lecture 25  (Monday 12/02/19). The first isomorphism theorem, the circle group R/Z and other examples.