## 330 Examples

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