By the end of this course, students will be able to:

• Recognize and analyze algebraic structures in both theoretical and applied contexts.
• Communicate mathematical reasoning and formal definitions clearly and effectively.
• Define and analyze group structures, including subgroups, normal subgroups, and quotient groups.
• Apply and prove the Isomorphism Theorems for groups.
• Explain and apply group actions and the Orbit-Stabilizer Theorem.
• Use the Sylow theorems to classify finite groups and analyze their subgroup structures.
• Understand the structure and properties of rings, subrings, ring homomorphisms, and ideals.
• Construct and analyze quotient rings and perform localization.
• Explain and identify properties of Euclidean domains, principal ideal domains (PIDs), and unique factorization domains (UFDs).