# 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 Representative.

### Equivalence, Congruence and Divisibility

• Use the Euclidean Algorithm to prove statements involving greatest common divisors.
• Compute the greatest common divisor and least common multiple of pairs of integers.
• 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.

### Groups: Lagrange’s Theorem and Isomorphism

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

### Permutation Groups

• Apply Sylow’s theorem to calculate Sylow subgroups.
• 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.

### Introduction to Groups

• Determine the group of symmetries of a given figure.
• 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.

### 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.