The following is a list of examples that I will be looking for. I
do really want at least one example for each of these. I will accept
no more than two examples for a given concept (unless indicated
otherwise). If you have an example of something other than these,
please tell me no later than in class before you want to present.
Probably, I will be happy to have examples of more things. Examples
from other sources count as your own examples, only examples from our book
count as book examples.
Section 2
Binary Operation (2.2-3)
Closed (and counterexample)
More Binary Operations (2.19-25) (including defined by tables)
Section 3
Isomorphic (and not) Binary Operations (3.8-3.10)
Structural properties (3.11ff)
Nonisomorphic via structural properties (3.15-17)
Section 4
Groups (4.2-14) (including via tables) [will take more than two examples]
First Problem Set [edits]
Section 5
Subgroups (5.6-13)
Subgroup Theorem (5.15-16)
Cyclic (Sub)groups (5.20-23)
Section 6
Division Algorithm
Subgroups of Finite Cyclic Groups (6.15,17)
Section 7
Generating Sets (7.1-2)
Intersection of subgroups
Cayley Graphs
Second Problem Set
First exam
Section 8
Composition of Permutations
Cayley's Theorem
Section 9
Orbits
Disjoint cycle notation
Even and odd permutations - composition of transpositions
Section 10
Cosets
Lagrange's Theorem
Index Theorem
Section 11
Direct Product of Groups
Cyclic Product of Cyclic Groups Theorem
Least Common Multiple and Order of Elements
Finitely Generated Abelian Groups
Applications Theorems
Third Problem Set [edits]
Section 13
Homomorphisms
Homomorphism properties Theorem
Kernel and Kernel Theorem
Normal subgroup
Section 14
Factor (quotient) groups
Factor groups from normal subgroups
Fundamental Homomorphism Theorem
Normal Subgroup Theorem
Conjugation and Automorphisms
Normal subgroups and factor groups.
Section 15
Simple Groups
Maximal normal subgroups.
Center
Commutator Subgroup
Fourth Problem Set
Second exam
Section 18
Ring - check theorem properties
Ring homomorphism
Ring isomorphism
Field
Subring
Subfield
Section 19
Zero divisors
Integral domains - and cancelation laws
Characteristic of a ring
Section 20
Fermat's Little Theorem
Subgroup of non-zero divisors
Euler's Theorem
Solving congruences
Fifth Problem Set [edits]
Section 21
Field of fractions
Section 22
Ring of Polynomials
Field of rational functions
Evaluation homomorphism
Zeroes of polynomials, rational, and algebraic.
Section 23
Polynomial Division Algorithm
Root/factor theorem
Irreducible Polynomials
Theorems for irreducibility
Eisenstein Criterion
Unique factorisation
(optional)
Section 26
Ring Homomorphism
Homomorphism Theorem.
Factor/quotient Rings
First Homomorphism Theorem for Rings
Quotient Rings
Section 27
Maximal Ideal
Prime ideal
Prime Field
Sixth Problem Set
Final exam