MTH 3340 - Abstract Algebra with Applications

• Prove elementary facts in number theory
• Solve linear congruence equations
• Determine if a given set with a binary operation is a group
• Determine if a given subset of a group is a subgroup
• Determine the possible orders of elements and subgroups of a given group
• Determine the cycle structure and orders of elements in a permutation group
• Determine if two groups are isomorphic; recognize standard classes of groups
• Determine the kernel and image of a group homomorphism; the cosets of a normal subgroup and the associated quotient group
• Describe the structure of a finite abelian group as a direct sum
• Use the RSA algorithm to encrypt and decrypt messages

• Basic set theory
• Functions and the algebra of functions including composition

• Direct and indirect proof methods
• Divisibility and prime numbers
• Modular arithmetic and linear congruence
• Euler’s formula and applications including k-th roots
• RSA public key cryptosystem
• Groups and subgroups, order of a group, cyclic groups, and Lagrange’s theorem
• Examples including symmetry groups, permutation groups, matrix groups
• Permutations, cycles, parity of permutations, alternating group
• Homomorphisms, normal subgroups, quotient groups
• Isomorphism and the isomorphism theorems
• Direct Products and structure theorems for cyclic and finitely generated abelian groups
• Group actions, orbits and stabilizers
• Other topics at instructor discretion: rings and fields, solvability of polynomials, impossibility of compass and straight edge constructions, or primality testing