Abstract Algebra I –

mth400

(3 credits)

This is the first course of a two-part course sequence presenting students with the applications of abstract algebraic theories. Students will investigate Group theory; including permutation groups, Abelian groups, finite groups, and homomorphism theorems.
This undergraduate-level course is 5 weeks. This course is available to take individually or as part of a degree or certificate program. To enroll, speak with an Enrollment Advisor.

Introduction to Groups

  • Determine if a given set, with respect to a given operation, forms a group.
  • Determine properties of permutations.
  • Decide if a given subset is a subgroup.
  • Determine the group of symmetries of a given figure.

Equivalence, Congruence and Divisibility

  • Verify properties of an equivalence relation.
  • Find representatives of equivalence classes.
  • Identify congruence classes modulo n.
  • Perform the division algorithm.
  • Apply group properties to Zn.
  • Use the Euclidean Algorithm to prove statements involving greatest common divisors.
  • Compute the greatest common divisor and least common multiple of pairs of integers.

Groups: Lagrange’s Theorem and Isomorphism

  • Determine subgroups for a given group and its elements’ order.
  • Identify generators and direct products of groups.
  • Determine the cosets of a given group.
  • Find all subgroups of a given group using Lagrange’s theorem and its corollaries.
  • Determine if given groups are isomorphic.
  • Use Cayley’s theorem to find permutations associated with an element of a group.

Homomorphisms

  • Determine if a given mapping is a homomorphism of groups.
  • Find a kernel of a homomorphism.
  • Identify normal subgroups for a given group and quotient groups.
  • Verify relationships between different groups using the Fundamental Homomorphism theorem.

Permutation Groups

  • Determine an action of a finite group on a set.
  • Use Burnside’s Counting theorem to determine the number of orbits for a group acting on a set.
  • Apply Sylow’s theorem to calculate Sylow subgroups.

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