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.
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.
Determine an action of a finite group on a set.
Verify relationships between different groups using the Fundamental Homomorphism theorem.
Identify normal subgroups for a given group and quotient groups.
Find a kernel of a homomorphism.
Determine if a given mapping is a homomorphism of groups.
Introduction to Groups
Determine if a given set, with respect to a given operation, forms a group.
Determine the group of symmetries of a given figure.
Decide if a given subset is a subgroup.
Determine properties of permutations.
Groups: Lagrange’s Theorem and Isomorphism
Use Cayley’s theorem to find permutations associated with an element of a group.
Determine if given groups are isomorphic.
Find all subgroups of a given group using Lagrange’s theorem and its corollaries.
Determine the cosets of a given group.
Identify generators and direct products of groups.
Determine subgroups for a given group and its elements’ order.
Equivalence, Congruence and Divisibility
Compute the greatest common divisor and least common multiple of pairs of integers.
Use the Euclidean Algorithm to prove statements involving greatest common divisors.
Apply group properties to Zn.
Perform the division algorithm.
Identify congruence classes modulo n.
Find representatives of equivalence classes.
Verify properties of an equivalence relation.
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