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.
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Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.