(due 10/28) Let \(G=\mathbb{Z}^3/\big(\mathbb{Z}(24,18,21)+\mathbb{Z}(27,-15,12)\big)\). Find a direct product of cyclic groups isomorphic to \(G\). §5.2: 2(a)(b)(c), 3(a)(b)(c), 13; §5.4: 10; §5.5: 7
Equivalence relations and partitions. Integers. Congruence.
Modular arithmetic. Definition of monoids and groups. Uniqueness of inverse. Examples.
The group of invertible elements in a monoid. Examples. Generalized Associativity Law. Conventions and notation.
Cancellation rules, order of an element. Dihedral group.
Euclidean space, orthogonal group. Linear representation of the dihedral group.
The group of permutations of a set. Elements of the dihedral group as permutations of the set of vertices. Two-line notation and composition of permutations.
Symmetric group: Cycles, length, transpositions. Cycle decomposition and its applications. Conjugating cycles.
Rings, commutative rings, division rings, fields. The ring of quaternions over the real numbers and over the integers. The quaternion group. Homomorphisms. Isomorphic groups.
Group actions and permutation representations. Left and right regular action, conjugation action. Subgroups, subgroup criterion.
Centralizer, center, and normalizer subgroups. Kernel of a homomorphism. Determinant and permutation matrices. Alternating group. Stabilizer subgroup. Kernel of an action.
The subgroup generated by a subset of a group. Cyclic groups. Order of powers of elements.
Cyclic groups: isomorphisms and subgroups. Lattice of subgroups.
Cosets. Quotient groups.
Lagrange's Theorem. Kernels of homomorphisms vs normal subgroups.
Isomorphism Theorems.
(Exam 1 Review)
Group Actions and Further Topics
Chapters 4–5, 6.3
G-sets and the Orbit-Stabilizer Theorem. Cayley's Theorem.
G acting on itself by conjugation. Class Equation. Center of a p-group is nontrivial.
Integer partitions and conjugacy classes of the symmetric group.
Automorphism groups, inner automorphisms, normalizer-mod-centralizer embeds in automorphism group of a subgroup.
Sylow's Theorem.
Sylow's Theorem (continued).
Applications of Sylow's Theorem.
Fundamental Theorem of Finitely Generated Abelian Groups.
Semidirect Products.
Application to the construction of non-abelian groups.
Free groups and groups given by a presentation.
(Exam 2 Review)
Ring Theory
Chapters 7–9
Definition of rings, examples, subrings.
Basic properties, integral domains and fields, monoid rings.
Subring generation, algebras, the Weyl algebra. Ring homomorphisms and ideals.
Quotient rings and the isomorphism theorems for rings.
Prime ideals and integral domains; maximal ideals and fields.
Universal property of group rings, augmentation ideal. Rings of fractions and their universal property.
The Remainder Theorem.
EDs are PIDs. Nonzero prime ideals in a PID are maximal.
PIDs are UFDs.
Irreducible elements in polynomial rings and quadratic integers.
Gauss' Lemma. R[x] is a UFD iff R is a UFD.
Eisenstein's Irreducibility Criterion.
(Final Exam Review)
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