mth400 | undergraduate

Abstract Algebra I

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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 To enroll, speak with an Enrollment Representative.

Course details:

Credits: 3
Duration: 5 weeks

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    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 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.
    • 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.

    Permutation Groups

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

    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.
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