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