Math 4032 Algebra I

Three credits. Three hours of lecture per week. Prerequisite: MATH 3151.

Introduction to group theory. Cosets and Lagrange’s theorem. Normal subgroups and quotient groups. Group homomorphisms. The isomorphism theorems. Finite groups. Permutation groups and Cayley’s theorem.

Justification

Group theory is a branch of algebra with applications to almost all branches of mathematics. A particular application is the study of symmetry — one of the central themes of modern mathematics. There are a multitude of applications of group theory in every branch of physics. In chemistry the classification of molecular spectra is based on the representation theory of groups. Accordingly this course is essential for any student of mathematics, physics or chemistry. It is of special interest to students of education who must study the algebraic properties of the number system.

Old Exams

Miscellaneous Notes

  1. Group axioms
  2. Subgroups
  3. Group homomorphisms
  4. Isomorphisms
  5. Cyclic Groups
  6. Lagrange’s Theorem
  7. Cayley’s Theorem
  8. Normal Subgroups
  9. First Isomorphism Theorem
  10. Greatest Common Divisor
  11. The Euclidean Group E(3)

Syllabus

  1. Preliminaries
    1. Functions and their properties.
    2. Binary and equivalence relations.
    3. Elementary number theory.
  2. Groups
    1. Definition of a group and discussion.
    2. Some special groups (finite, abelian, cyclic, etc.)
    3. Examples of Groups.
    4. Simple properties of groups.
  3. Subgroups
    1. Definition of a subgroup and discussion.
    2. Examples of subgroups.
    3. Simple properties of subgroups.
    4. A subgroup criterion.
    5. Some special subgroups (a cyclic subgroup, a subgroup generated by a set of elements, the stabilizer of an element in a permutation group, the centralizer, the center of a group).
    6. Groups having non trivial subgroups.
    7. A finite subgroup criterion.
  4. A Table of a Group and its Properties.
  5. Cosets
    1. Definition of a coset and discussion.
    2. Examples of a coset
    3. A coset as an equivalence class of an equivalence relation.
  6. Lagrange’s Theorem and Corollaries.
  7. Normal Subgroups
    1. Definition of a normal subgroup and discussion.
    2. Examples of normal subgroups.
    3. Properties of normal subgroups.
  8. Factor Groups
    1. Cosets of a normal subgroup and their properties.
    2. A binary operation on the cosets of a normal subgroup induced by the group operation.
    3. A definition of a factor group, discussion and examples.
  9. Homomorphism and Isomorphism of Groups
    1. Definition of a homomorphism and an isomorphism of groups and discussion.
    2. Examples of homomorphisms.
    3. Properties of homomorphisms.
    4. The kernel of a homomorphism and its properties.
    5. Cosets of the kernel and the inverse image of elements under the homomorphism.
    6. Homomorphism Theorem and its Applications.

Bibliography

  1. I. Herstein, Abstract Algebra, MacMillan Publishing Company, 1990.
  2. J.B. Fraleigh with Historical Notes by Victor Katz. A First Course in Abstract Algebra, (5th edition), Addison-Wesley, 1994.
  3. D. Maclane and G. Birkhoff, Algebra. 3rd Ed., Chelsea Pubishing Company, New York, N. Y., 1988.