Sets, their unions, intersections, differences, direct (or cartesian) products. Subsets. Maps between sets, injective, surjective and bijective maps. Images and preimages of subsets. Composition of maps. Identity map and Inverse of map.
Binary operations on sets. Associativity, multiplicativity. Identity and inverse elements with respect to a binary operation. Groups, semigroups, monoids. Cayley table of a group. Direct products of groups. Subgroups. Intersections of subroups. Generators of a subgroup.
Permutation group of a set (the group of all bijective self-maps). Symmetric group Sn. Parity (sign) of a permutation, even and odd permutations. Alternating subgroup An of Sn. Group of Isometries. Matrix groups GLn, SLn, On, SOn, Un, SUn.
Integer division with remainder. Additive subgroups of Z. Greatest common divisor. Euclidean algorithm. Unique prime factorization. Binary relations, equivalence relations, partitions. Congruence relation and classes of integers modulo n. The set of congruence classes Zn modulo n as additive group and multiplicative monoid. Multiplicative group Zn*.
Cosets of a subgroup in a group. Lagrange's Theorem.
Group homomorphisms and isomorphisms. Kernel of homomorphism. Normal subgroup. Quotient group modulo normal subgroup. First isomorphism theorem.
Chinese remainder theorem. Group actions on a set. Stabilizers and orbits of a group action. Sylow's Theorem. Classification of finite abelian groups.
For exam-related problems look in TCD past examination papers and Mathematics department examination papers.
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