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.
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.
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.
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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