MATH 531: Some topics we covered in Chapters 0-4

• Induction & the well-ordering principle
• The Division Algorithm
• The g.c.d. (definition; expression as linear combination; applications)
• The Fundamental Theorem of Arithmetic (unique factorization)
• Arithmetic mod n ($\equiv\pmod{n},\oplus_n,\odot_n, \mathbb {Z} _n$)
• Equivalence relations (definition, equivalence classes, partitions)
• Functions (1-1, onto, bijective; composition; inverses)
• Symmetry of plane figures & the Dihedral groups D_n
• Groups (definition, elementary properties, order of a group, Abelian groups)
• Examples of groups (many, including $\mathbb {Z}$, $( \mathbb {R} _{\gt},\cdot)$,$\mathbb {Z} _n$, U(n), GL₂, SL₂, D_n)
• Subgroups (definition, ways to verify, examples)
• The center of a group; the centralizer of an element
• The order of an element a; when aⁱ and a^j are equal to each other and when they equal e
• The cyclic (sub)group $\langle{a}\rangle$; if |a|=n, then $\langle{a}\rangle=\{e,a,\dots,a^{n-1}\}$
• The Fundamental Theorem of Cyclic Groups (information about the subgroups of a cyclic group)