MATH 531: Some topics we covered in Chapters 5-11

• Permutations - notation & operations
• Cycles; the cycle decomposition of a permutation & its uses
• Transpositions; even & odd permutations; the subgroup A_n
• Isomorphism (the relation) of groups; it's an equivalence relation
• Isomorphisms (functions) & homomorphisms
• Properties preserved by isomorphism (Abelian, cyclic, orders of elements)
• Determining whether groups are isomorphic
• Automorphisms, including conjugation
• Cosets & their properties
• Lagrange's Theorem (the order of a subgroup divides the order of the group)
• Applications (groups of prime order, Fermat's Little Theorem, Euler's Theorem)
• Normal subgroups (pp. 172-173): characterizations via cosets & conjugation
• Direct product of groups (pp. 150-154)
• When is a direct product cyclic?
• Fundamental Theorem of finite Abelian groups (pp. 211-213)