Dr. Bob's Index of Class Notes

This webpage contains links to topics covered in Robert "Dr. Bob" Gardner's online class notes. The topics are listed alphabetically and links are labeled by course number and section number.

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

A
abelian group characterizations 5410 II.1 page 2 Theorem II.1.1
absolute convergence of a series definition 5510 III.1 page 1 Definition
absolute convergence of a series implies convergence 5510 III.1 page 1 Proposition III.1.1
action of a group of order p^n for some prime p 5410 II.5 page 2 Lemma II.5.1
action of group G on set S definition 5410 II.4 page 1 Definition II.4.1
Addition of Complex Numbers 5510 I.2 page 1 (Definition)
algebraically closed division ring definition 5410 Quaternions Algebraic Supplement page 18 Definition (Lam, page 169).
algebraically closed field definition 5410 Quaternions Algebraic Supplement page 17 Note.
alternating group is generated by 3-cycles 5410 I.6 page 8 Lemma I.6.11
alternating group is simple for n ≠ 4 5410 I.6 page 8 Theorem I.6.10
alternating group on n letters definition 5410 I.6 page 7 Theorem I.6.8
any two coproducts for a given family of objects are equiavelent 5410 I.7 page 8 Theorem I.7.5
any two products for a given family of objects are equivalent 5410 I.7 page 7 Thorem I.7.3
any two universal objects are equivalent 5410 I.7 page 11 I.7.10
Argument of a Complex Number 5510 I.4 page 1 (Definition)
arithmetic involving left and right ideals in a ring 5410 III.2 page 4 Theorem III.2.6
ascending central series of G definition 5410 II.7 page 2 Definition II.7.1
ascending chain condition (ACC) 5410 II.3 page 3 Definition II.3.2
associates definition 5410 III.3 page 1 Definition III.3.1
associates of irreducible elements are irreducible 5410 III.3 page 3 Theorem III.3.4
associates relation defines an equivalence relation 5410 III.3 page 1 Theorem III.3.2
Automorphism 5410 I.2 page 1 (Definition I.2.1)
automorphism of rings definition 5410 III.1 page 7 Note
Axiom of Completeness Characterization 5510 I.1 page 1 (Note)

B
bases of same free abelian group have same cardinality 5410 II.1 page 3 Theorem II.1.2
basis of an abelian group definition 5410 II.1 page 1 Definition
Betti number of a group definition 5410 II.2 page 4 Note
bicyclic permutation defintion 5410 I.6 page 2 Note.
binary algebraic structure 5410 I.1 page 2
binary operation 5410 I.1 page 1
binary operation in free group definition 5410 I.9 page 3 Definition
Binomial theorem 5410 III.1 page 6 Theorem III.1.6
Boundary of a set definition 5510 II.1 page 8 (Definition II.1.12)
Boundary of a set properties 5510 II.1 page 8 (Theorem II.1.13)
Butterfly Lemma 5410 II.8 page 5 Lemma II.8.9

C
ℂ is complete 5510 II.3 page 3 (Proposition II.3.6)
ℂ is not an ordered field 5510 Ordering-ℂ page 2 (Theorem 3)
cancellation implies no zero divisors 5410 III.1 page 2 Lemma.
canonical epimorphism associated with the internal direct product 5410 II.3 page 7 Definition.
canonical epimorphism (see also canonical homomorphism) 5410 I.5 page 4 (Definition)
canonical homomorphism (see also canonical epimorphism) 5410 I.5 page 4 (Definition)
canonical injection definition 5410 I.8 page 3 Definition.
canonical project of the direct product definition 5410 I.8 page 2 Theorem I.8.1
Cantor's Theorem 5510 II.3 page 4 (Cantor's Theorem)
cardinal number of set element is index of stabilizer 5410 II.4 page 3 Theorem II.4.3
category definition 5410 I.7 page 2 Definition I.7.1
Cauchy sequence definition 5510 II.3 page 2 (Definition II.3.5)
Cauchy Sequences Theorem 5510 I.3 page 2 (Theorem I.3.B)
Cayley's Theorem 5410 II.4 page 4 Corollary II.4.6
center of nontirival finite p-group has more than one element 5410 II.5 page 4 Corollary II.5.4
centralizer of x definition 5410 II.4 page 3 Definition.
centralizer of x in subgroup H definition 5410 II.4 page 3 Definition.
chain condition implies characterizatons of automoprhisms 5410 II.3 page 5 Lemma II.3.4
chain of ideals in principal ideal rings eventually stop 5410 III.3 page 5 Lemma III.3.6
characterization of irreducible elements 5410 III.3 page 2 Theorem III.3.4
characterization of prime elements 5410 III.3 page 2 Theorem III.3.4
characterizations of division and associates 5410 III.3 page 1 Theorem III.3.2
characteristic of a ring definition 5410 III.1 page 7 Definition III.1.8
characteristic of a ring with unity can be found by considering unity only 5410 III.1 page 7 Theorem III.1.9 (ii)
characteristic subgroup definition 5410 II.7 page 8 Definition.
characteristic subgroup definition 5410 Semidirect Product Supplement page 2 Proposition DF.5.7 (3)
characteristic subgroup of a normal subgroup is normal in group G 5410 II.7 page 9 Lemma II.7.13
characteristic zero definition 5410 III.1 page 7 Definition III.1.8
characterization of a complete metric subspace 5510 II.3 page 4 (Proposition II.3.8)
characterization of isomorphic groups in terms of the quotient group of the torsion subgroups of each group 5410 II.2 page 5 Corollary II.2.7
characterization of prime ideal in commutative rings with identity 5410 III.2 page 9 Theorem III.2.16
characterization of prime ideals in commutative rings 5410 III.2 page 8 Theorem III.2.15
Chinese Remainder Theorem 5410 III.2 page 13 Theorem III.2.25 Chinese Remainder Theorem
class equation of group G definition 5410 II.4 page 4 Definition.
classification of groups of order pq where p and q are primes 5410 II.6 page 1 Proposition II.6.1
classification of prime ideals 5410 III.2 page Exercise III.2.14
Closed Ball 5510 II.1 page 1 (Definition)
Closed definition 5510 II.1 page 6 (Definition II.1.10)
closed set contains all its limit points 5510 II.3 page 1 (Proposition II.3.2, Proposition II.3.4)
Closure of a set definition 5510 II.1 page 8 (Definition II.1.12)
Closure of a set properties 5510 II.1 page 8 (Theorem II.1.13)
Closure of Binary Operation 5410 I.2 page 4 (Definition I.2.4)
closure of set contains all its limit points 5510 II.3 page 2 (Proposition II.3.4)
commutative diagram 5410 I.5 page 4 (Note)
commutative ring definition 5410 III.1 page 1 Definition III.1.1
commutator of x and y definition 5410 Semidirect Product Supplement page 1 Definition. (1)
commutator subgroup is normal 5410 II.7 page 4 Theorem II.7.8
commutator subgroup of G 5410 II.7 page 4 Definition II.7.7
commutator subgroup of G definition 5410 Semidirect Product Supplement page 1 Definition. (3)
compact definition 5510 II.4 page 1 (Definition II.4.1)
Compactification definition 5510 Extended Complex Plane page 4 Definition
Compactness of ℂ∞ 5510 Extended Complex Plane page 5 Compactness of ℂ∞ Theorem
complement of a subgroup definition 5410 II.3 page 1 Definition
complete direct sum definition (see also direct product of groups) 5410 I.8 page 1 Definition
Completeness of Ordered Field 5510 I.1 page 1 (Recall)
completness of metric space definition 5510 II.3 page 2 (Definition II.3.5)
Complex Numbers Definition 5510 I.2 page 1 (Definition)
component of a metric space 5510 II.2 page 4 (Definition II.2.5)
composite definition 5410 I.7 page 2 Definition I.7.1 (ii)
composition series definition 5410 II.8 page 2 Definition II.8.3
compositions of continuous functions are continuous 5510 II.5 page 2 Proposition II.5.5
concrete category definition 5410 I.7 page 8 Definition I.7.6
conditions under which there is a ring isomorphism 5410 III.2 page 12 Theorem III.2.24
conditions under which the system of congruences has an intgeral solution that is uniquely determined modulo m 5410 III.2 page 14 Corollary III.2.26
congruence relation on a monoid 5410 I.1 page 8 (Theorem I.1.5)
congruent modulo an ideal definition 5410 III.2 page 13 Definition.
conjugacy class of group element x results 5410 II.4 page 4 Corollary II.4.4
conjugacy class of x is orbit of x if group G acts on itself by conjugation 5410 II.4 page 3 Definition.
conjugate of a quaternion definition 5410 Quaternions Algebraic Supplement page 15 Note.
Conjugate of Complex Number 5510 I.2 page 2 (Definition)
conjugate of set element x definition 5410 II.4 page 2 Example.
conjugation by group element definition 5410 II.4 page 2 Example.
conjugation is an equivalence relation in H 5410 Quaternions Algebraic Supplement page 15 Note.
conjugtaion by group element is an automorphism 5410 II.4 page 5 Corollary II.4.7
connected characterization 5510 II.2 page 3 (Theorem II.2.3)
connected metric space 5510 II.2 page 2 (Definition II.2.1)
connected subset 5510 II.2 page 2 (Definition II.2.1)
continuity alone does not imply uniformly continuous 5510 Lipschitz Primer page 4 Example 1
continuous at a point characterization 5510 II.5 page 1 Proposition II.5.2
continuous at a point definition 5510 II.5 page 1 Definition II.5.1
continuous function definition 5510 II.5 page 1 Definition II.5.1
continuous function properties 5510 II.5 page 6 Theorem II.5.8
continuous functions on compact sets are uniformly continuous 5510 II.5 page 8 Theorem II.5.15
continuously differentiable function of a complex variable has a power series representation 5510 Lipschitz Primer page 7 Note
continuous on a set characterization 5510 II.5 page 2 Proposition II.5.3
continuous on a set topological definition 5510 II.5 page 3 Definition
continuous on compact set implies uniformly continuous 5510 Lipschitz Primer page 4 Thorem 5
conugate of product of quaternionics is product of conjugates in reverse order 5410 Quaternions Algebraic Supplement page 16 Lemma.
convergence of a sequence in a metric space definition 5510 II.3 page 1 (Definition II.3.1)
Convergence of a sequence of complex numbers 5510 Extended Complex Plane page 4 Sequences in ℂ∞ Theorem
convergence of a series definition 5510 III.1 page 1 Definition
coproduct definition 5410 I.7 page 7 Definition.
correspondence of normal subgroups 5410 I.5 page 9 (Theorem I.5.11)
couniversal object definition (see also termainl object) 5410 I.7 page 10 Definition I.7.9
countable number of components of a open subset of ℂ 5510 II.2 page 5 (Theorem II.2.9)
cycle of length r definition 5410 I.6 page 1 Definition I.6.1
Cyclic Group 5410 I.2 page 5 (Definition)

D
decomposition of metric space into components 5510 II.2 page 4 (Theorem II.2.7)
De Moivre's Formula 5510 I.4 page 2 (Corollary)
Dense definition 5510 II.1 page 9 (Definition II.1.14)
derivative at a point definition 5510 Lipschitz Primer page 2 Definition
descending chain condition (DCC) 5410 II.3 page 3 Definition II.3.2
diameter of a set 5510 II.3 page 3 (Definition)
dicyclic group definition 5410 II.6 page 2 Note.
differentiable implies Lipschitz if and only if derivative is bounded 5510 Lipschitz Primer page 3 Theorem 2
differentiable implies locally Lipschitz 5510 Lipschitz Primer page 2 Theorem 1
differentiable on set X definition 5510 Lipschitz Primer page 2 Definition
dihedral group of degree n definition 5410 I.6 page 8 Definition
direct product of a family of rings 5410 III.2 page 11 Definition.
direct product of a family of rings is a product in the category of rings 5410 III.2 page 12 Theorem III.2.23
direct product of a finite number of nilpotent groups is nilpotent 5410 II.7 page 3 Proposition II.7.3
direct product of groups 5410 I.1 page 7
direct product of groups definition (see also complete direct sum of groups) 5410 I.8 page 1 Definition
direct product of groups is a group 5410 I.8 page 2 Theorem I.8.1
direct product of groups is a product in the category of abelian groups 5410 I.8 page 2 Note
direct product of groups is a product in the category of groups 5410 I.8 page 2 Theorem I.8.2
direct product of ideals definition 5410 III.2 page 12 Definition.
direct sum of groups 5410 I.1 page 7
disjoint permutations definition 5410 I.6 page 1 Definition I.6.2
distance from a point to a set definition 5510 II.5 page 5 Definition
distance from a point to a set properties 5510 II.5 page 5 Definition
distance from set A to set B definition 5510 II.5 page 8 Definition II.5.16
distance from set A to set B property 5510 II.5 page 9 Theorem II.5.17
Distance in ℂ 5510 I.6 page 2 (Definition)
divides definition 5410 III.3 page 1 Definition III.3.1
division ring characterization 5410 III.1 page 3 Note
division ring definition 5410 III.1 page 3 Definition III.1.5
division ring has no proper left or right ideals 5410 III.2 page 2 Note.
division ring of real quaternions definition (see also real quaternions division ring) 5410 III.1 page 5 Example
domain of Gaussian integers definition 5410 III.3 page 6 Example.
D_n is solvable 5410 II.8 page 3 Example.
D_n not nilpotent when n is odd 5410 II.7 page 2 Example 1.

E
elementary divisors definition 5410 II.2 page 4 Definition
empty word definition 5410 I.9 page 1 Definition
Endomorphism 5410 I.2 page 1 (Definition I.2.1)
Epimorphism 5410 I.2 page 1 (Definition I.2.1)
epimorphism of rings definition 5410 III.1 page 7 Note
Equation of a line in ℂ 5510 I.5 page 1 (Recall)
equivalence class of element under right and left congruence modulo H definition 5410 I.4 page 1 (Theorem I.4.2)
equivalence definition 5410 I.7 page 3 Definition.
equivalent conditions ona commutative ring with identity 5410 III.2 page 10 Corollary III.2.21
equivalent, definition of 5410 I.7 page 3 Definition.
equivalent subnormal series definition 5410 II.8 page 4 Definition II.8.7
euclidean domain definition 5410 III.3 page 5 Definition III.3.8
euclidean ring definition 5410 III.3 page 5 Definition III.3.8
evaluation of an element of H[q] at an element is not a homomorphism 5410 Quaternions Algebraic Supplement page 10 Note.
even permutation definition 5410 I.6 page 3 Definition I.6.6
every abelian group is nilpotent 5410 II.7 page 2 Note.
every abelian group is solvable 5410 II.7 page 6 Note.
every abelian group is the homomorphic image of a free abelian group 5410 II.1 page 5 Theorem II.1.4
every characteristic subgroup is normal 5410 II.7 page 8 Note.
every conjugate of a sylow p-subgroup is a sylow p-subgroup 5410 II.5 page 5 Corollary II.5.8 (ii)
every euclidean domain is a unique factorization domain 5410 III.3 page 6 Theorem III.3.9
every euclidean ring is a principal ideal ring with identity 5410 III.3 page 6 Theorem III.3.9
every field is a unique factorization domain 5410 III.3 page 3 Note.
every field is an integral domain 5410 III.1 page 3 Note
every finite abelian group has a subgroup for each factor of its order 5410 II.2 page 2 corollary II.2.4
every finite cyclic group is isomorphic to a sum of finite cyclic groups 5410 II.2 page 1 Lemma II.2.3
every finite cyclic group of order m is isomorphic to additive group Z 5410 I.3 page 1 (Theorem I.3.2)
every finite group G has a composition series 5410 II.8 page 3 Theorem II.8.4
every finite group G has a nontrivial sylow p-subgroup 5410 II.5 page 5 Note.
every finite group is isomorphic to a unique direct product of a finite number of indecomposable subgoups 5410 II.3 page 8 Note.
every finitely generated abelian group is isomorphic to a direct sum of cyclic groups 5410 II.2 page 1 theorem II.2.1, theorem II.2.2
every finite p-group is nilpotent 5410 II.7 page 3 Proposition II.7.2
every group G is isomorphic to a quotient group of a free group (see also Gallian's "Universal Quotien Group Property") 5410 I.9 page 5 Thorem
every group G is the homomorphic image of a free group 5410 I.9 page 5 Corollary I.9.3
every group of order 2p is isomorphic to Z_2p or D_p for odd p 5410 II.6 page 1 Corollary II.6.2
every group of order 483 is simple 5410 II.5 page 8 Example.
every ideal in R/I is of the form J/I where J is an ideal which contains I 5410 III.2 page 7 Theorem III.2.13
every infinite cyclic group is isomorphic to additive group Z 5410 I.3 page 1 (Theorem I.3.2)
every maximal ideal is prime in a commutative ring with identity 5410 III.2 page 10 Theorem III.2.19
every nilpotent group is solvable 5410 II.7 page 7 Proposition II.7.10
every normal sylow p-subgroup of G is fully invariant 5410 II.7 page 9 Lemma II.7.13
every permutation can be written as a product of tanspositions 5410 I.6 page 3 Corollary I.6.5
every permutation is a product of disjoint cycle 5410 I.6 page 2 Theorem I.6.3
every refinement of a solvable series is a solvable series 5410 II.8 page 3 Theorem II.8.4
every ring has at least two ideals, the trivial ideal {0} and itself 5410 III.2 page 1 Examples.
every ring may be embeded into another ring with either same characteristic or characteristic zero 5410 III.1 page 8 Theorem III.1.10
every subgroup of a cyclic group is cyclic 5410 I.3 page 2 (Theorem I.3.5)
existence of homomorphism from G to Aut(G) with kernel C(G) 5410 II.4 page 5 Corollary II.4.7
exponential function definition 5510 III.1 page 4 Definition
Extended complex plane 5510 I.6 page 1 (Note)
external direct product of a family of rings 5410 III.2 page 11 Definition.
external direct product of a family of rings is a product in the category of rings 5410 III.2 page 12 Theorem III.2.23
external direct product of normal subgroups definition 5410 Semidirect Product Supplement page 3 Definition.i
external direct sum definition 5410 I.8 page 2 Definition I.8.3
external direct sum of abelian groups is a coproduct in the category of abelian groups 5410 I.8 page 3 Theorem I.8.5
external weak direct product definition 5410 I.8 page 2 Definition I.8.3
external weak direct product is a normal subgroup of the direct product of groups 5410 I.8 page 3 Theorem I.8.4
Extreme Value Theorem 5510 II.5 page 7 Corollary II.5.12 Extreme Value Theorem
Extreme Value Theorem in the complex setting 5510 II.5 page 8 Corollary II.5.13, Corollary II.5.14

F
actor group (see also quotient group) 5410 I.5 page 3 (Theorem I.5.4, Definition, Note)
factor theorem 5410 Quaternions Algebraic Supplement page 9 The Factor Theorem (Hungerford's Theorem III.6.6)
factor theorem in a ring with unity 5410 Quaternions Algebraic Supplement page 13 Proposition 16.2 of Lam. (The Factor Theorem in a Ring with Unity).
f + g notation 5410 II.3 page 6 Notation.
field definition 5410 III.1 page 3 Definition III.1.5
finite group G is solvable if and only if G has a composition series whose factors are cyclic and of prime order 5410 II.8 page 4 Proposition II.8.6
finite intersection property 5510 II.4 page 3 (Definition)
finitely generated ideal generated by a set X definition 5410 III.2 page 3 Definition III.2.4
Finitely Generated Subgroup 5410 I.2 page 5 (Definition)
finite nilpotent group has subgroups of order a divisor of the order of the group 5410 II.7 page 4 Corollary II.7.6
finite product of nonempty subsets of a ring definition 5410 III.2 page 4 definition
finite sum of normal nilpotent epimorphisms is nilpotent 5410 II.3 page 6 Corollary II.3.7
finite sums of nonempty subsets of a ring definition 5410 III.2 page 4 Definition
First Isomorphism Theorem 5410 I.5 page 5 (Corollary I.5.7)
First Isomorphism Theorem 5410 III.2 page 6 Corollary III.2.10. First Isomorphism Theorem
First Sylow Theorem 5410 II.5 page 5 Theorem II.5.7
Fitting's Lemma 5410 II.3 page 5 Lemma II.3.5 (Fitting's Lemma.)
free abelian group on the set X definition 5410 II.1 page 2 Definition
free group F is a free object on the set X in the category of groups 5410 I.9 page 5 Theorem I.9.2
free group on set X definition 5410 I.9 page 4 Theorem I.9.1
free group properties 5410 I.9 page 4 Note
free on the set X definition 5410 I.7 page 9 Definition I.7.7
free product of a family of groups definition 5410 I.9 page 12 "Definition."
free product of a family of groups is a coproduct in the category of groups 5410 I.9 page 13 Theorem I.9.6
f=u--lim(f_n) definiton 5510 II.6 page 1 Definition
fully invariant definition 5410 II.7 page 8 Definition.
fully invariant subgroup implies characteristic subgroup 5410 II.7 page 8 Note.
Fundamental Homomorphism Theorem 5410 I.5 page 4 (Theorem I.5.6)
Fundamental Theorem of Algebra for Quaternions 5410 Quaternions Algebraic Supplement page 18 Theorem 16.14 ("Niven-Jacobson" in Lam) Fundamental Theorem of Algebra for Quaternions
fundamental theorem of finitely generated abelian groups 5410 II.2 page 3 Theorem II.2.6
Fundamental Theorem of Free Abelian Groups 5410 II.1 page 4 Proposition II.1.3

G
Gallian's "Universal Quotient Group Property" 5410 I.9 page 5 Theorem
Gauss-Lucas Theorem 5510 Ilieff-Sendov page 1 (Theorem 1)
Generalized Associative Law in a semigroup 5410 I.1 page 10 (Theorem I.1.6)
Generalized Commutative Law in a commutative semigroup 5410 I.1 page 11 (Corollary I.1.7)
generators of an ideal generated by a set X defintion 5410 III.2 page 3 Definition III.2.4
generators of cyclic groups 5410 I.3 page 2 (Theorem I.3.6)
generators of dihedral group of degree n characterization 5410 I.6 page 9 Theorem I.6.13
Generators of Subgroup 〈X〉 5410 I.2 page 5 (Definition)
geometric series definition 5510 III.1 page 3 Definition
G indecomposable and satisifies ACC and DCC with f a normal endomorphism, then f is nilpotent or automorphism 5410 II.3 page 6 Corollary II.3.6
G is isomorphic to subgroup of Sn sufficient conditions 5410 II.4 page 6 Corollary II.4.9
greatest common divisor in different settings 5410 III.3 page 7 Theorem III.3.11
greatest common divisor of a a subset of a commutative ring definition 5410 III.3 page 6 Definition III.3.10
greatest common divisors may not be unique for a given set in a ring 5410 III.3 page 6 Note.
greatest common divisors may not exist for a given set in a ring 5410 III.3 page 6 Note.
group action induces a homomorphism 5410 II.4 page 4 Theorem II.4.5
group 5410 I.1 page 2 (Definition I.1.1)
group element properties 5410 I.3 page 2 (Theorem I.3.4)
Group Generated by Subgroups 5410 I.2 page 6 (Definition)
group G is an internal weak direct product of a collection of subgroups, conditions under which 5410 I.8 page 6 Note.
group is isomorphic to the weak direct product of a family of its subgroups, conditions under which a 5410 I.8 page 4 Theorem I.8.6
group of order multiple a prime p has element of order p 5410 II.5 page 2 Theorem II.5.2
group of rationals modulo 1 (Prüfer group) 5410 I.1 page 8
group ring of G over R definition 5410 III.1 page 4 Example
group ring of G over R is a group under addition and multiplication 5410 III.1 page 4 Example
group ring of G over R is commutative if and only if both R and G are commutative 5410 III.1 page 5 Example
group satisfying a chain condition on normal subgroups is isomorphic to a direct product of finite number of indecomposable subgroups 5410 II.3 page 4 Theorem II.3.3
G satisfying ACC and DCC is indecomposable if and only if any normal endomorphism is either nilpotent or automorphism 5410 II.3 page 6 Note.

H
Half-plane in ℂ 5510 I.5 page 1,2 (Note)
H a proper subgroup of a nilpotent group G implies H is a proper subgroup of its normalizer 5410 II.7 page 3 Lemma II.7.4
Heine-Borel Theorem 5510 II.4 page 6 (Theorem II.4.10)
H is a sylow p-subgroup of G if and only if order of H is a power of a prime 5410 II.5 page 5 Corollary II.5.8 (i)
homeomorphism between topological spaces definition 5510 II.4 page 7 (Definition)
homomorphic images of solvable groups are solvbable 5410 II.7 page 7 Theorem II.7.11 (i)
Homomorphism 5410 I.2 page 1 (Definition I.2.1)
homomorphism of rings definition 5410 III.1 page 6 Definition III.1.7
homomorphism of rings induces a homomorphism of quotient rings 5410 III.2 page 6 Corollary III.2.11
homomorphism of rings may not map identities to identities 5410 III.1 page 7 Note.

I
ideals in different settings 5410 III.2 page 3 Theorem III.2.5
ideal subring definition 5410 III.2 page 1 Definition III.2.1
Identities 5510 I.2 page 2 (Theorem I.2.A, Corollary I.2.A)
identity map between semidirect product and external direct product is a group homomorphism 5410 Semidirect Product Supplement page 7 Proposition DF.5.11
Ilieff-Sendov Conjecture 5510 Ilieff-Sendov page 5 (Conjecture)
Image 5410 I.2 page 2 (Definition I.2.2)
image of a homomorphism of rings is a subring 5410 III.2 page 1 Examples.
indecomposable group definition 5410 II.3 page 1 Definition II.3.1
index of a subgroup 5410 I.4 page 2 (Definition I.4.4)
index of intersection of subgroups 5410 I.4 page 4 (Proposition I.4.8, Proposition I.4.9)
index of p-subgroup in its normalizer is congruent to index of the p-subgroup in the group modulo p 5410 II.5 page 4 Lemma II.5.5
index of p-subgroup result 5410 II.5 page 4 Corollary II.5.6
index of subgroup of subgroup 5410 I.4 page 3 (Theorem I.4.5)
initial object definition (see also universal object) 5410 I.7 page 10 Definition I.7.9
inner automorphism induced by g definition 5410 II.4 page 5 Definition.
Inner product 5510 II.1 page 3 (Example)
integers are a unique factorization domain 5410 III.3 page 4 Note.
integral domain definition 5410 III.1 page 3 Definition III.1.5
Interior of a set definition 5510 II.1 page 8 (Definition II.1.12)
Interior of a set properties 5510 II.1 page 8 (Theorem II.1.13)
Intermediate Value Theorem 5510 II.5 page 7 Theorem II.5.11 Intermediate Value Theorem
internal direct product of ideals contains only isomorphic images of the ideals in the product 5410 III.2 page 12 Note.
internal direct product of ideals definition 5410 III.2 page 12 Definition.
internal direct product of normal subgroups definition 5410 Semidirect Product Supplement page 3 Definition.
internal direct product of subgroups is a subgroup characterization 5410 Semidirect Product Supplement page 5 Note.
internal direct product of subgroups is a subgroup sufficient conditions 5410 Semidirect Product Supplement page 5 Corollary DF.3.15
internal direct sum definition (see also internal weak direct product) 5410 I.8 page 5 Definition I.8.8
internal weak direct product definition (see also internal direct sum) 5410 I.8 page 5 Definition I.8.8
intersection of left ideals is a left ideal 5410 III.2 page 2 Corollary III.2.3
intersection of right ideals is a right ideal 5410 III.2 page 2 Corollary III.2.3
Intersection of Subgroups is a Subgroup 5410 I.2 page 5 (Corollary I.2.6)
invariant factors definition 5410 II.2 page 4 Definition
Inverse Image 5410 I.2 page 2 (Definition I.2.2)
invertible element definition 5410 III.1 page 3 Definition III.1.4
involution definition 5410 I.6 page 2 Note.
irreducible and prime elements coincide in unique factorization domains 5410 III.3 page 4 Note.
irreducible ring element definition 5410 III.3 page 2 Definition III.3.3
Isomoprhism, Characterization 5410 I.2 page 3 (Theorem I.2.3)
Isomorphism 5410 I.2 page 1 (Definition I.2.1)
isomorphism of rings definition 5410 III.1 page 7 Note
isotropy group of x definition 5410 II.4 page 2 Theorem II.4.2 ii
i^th derived subgroup of G definition 5410 II.7 page 5 Definition.

J
Join of Subgroups 5410 I.2 page 6 (Definition)
Jordan-Holder Theorem 5410 II.8 page 6 Theorem II.8.11

K
Kernel 5410 I.2 page 2 (Definition I.2.2)
kernel of a homomorphism of rings definition 5410 III.1 page 6 Definition III.1.7
kernel of a homomorphism of rings is an ideal 5410 III.2 page 6 Theorem III.2.8
kernel of homomorphism of groups is a normal subgroup 5410 I.5 page 4 (Theorem I.5.5)
Krull-Schmidt Theorem 5410 II.3 page 8 Theorem II.3.8 (The Krull-Schmidt Theorem)

L
Lagrange's Theorem 5410 I.4 page 3 (Corollary I.4.6 Lagrange's Theorem)
Lebesgue's Covering Lemma 5510 II.4 page 4 (Lemma II.4.8)
left and right inverses in rings with identity coincide 5410 III.1 page 3 Note
left ideal generated by a set X definition 5410 III.2 page 3 Definition III.2.4
left ideal subring defintiion 5410 III.2 page 1 Definition III.2.1
left inverse definition 5410 III.1 page 3 Definition III.1.4
left invertible ring element definition 5410 III.1 page 3 Definition III.1.4
left or right nonzero ideal of a ring is porper if and only if it contains no units of the ring 5410 III.2 page 2 Note.
left root definition 5410 Quaternions Algebraic Supplement page 13 Definition 16.1 of Lam.
left translation definition 5410 II.4 page 2 Example.
left zero divisor definition 5410 III.1 page 2 Definition III.1.3
Lexicographic ordering 5510 Ordering-C page 3 (Note)
limit definition in a topological space 5510 II.4 page 6 (Definition)
limit infimum always exists 5510 III.1 page 2 Note
limit infimum as smallest subsequential limit point 5510 III.1 page 2 Note
limit infimum definition 5510 III.1 page 1 Definition
limit of a function in a topological space definition 5510 II.5 page 3 Definition
limit of a sequence exists if and only if limit supremum equals limit infimum 5510 III.1 page 2 Note
limit of a sequence of continuous functions is continuous 5510 II.6 page 1 Theorem II.6.1
limit point definition 5510 II.3 page 2 (Definition II.3.3)
limit supremum always exists 5510 III.1 page 2 Note
limit supremum as largest subsequential limit point 5510 III.1 page 2 Note
limit supremum definition 5510 III.1 page 1 Definition
linear combinations of continuous functions are continuous 5510 II.5 page 2 Proposition II.5.4
line segment between two points 5510 II.2 page 2 (Definition)
Lipschitz constant definition 5510 Lipschitz Primer page 2 Definition
Lipschitz function definition 5510 II.5 page 4 Definition II.5.6
Lipschitz functions definition 5510 Lipschitz Primer page 6 Note (second paragraph)
Lipschitz implies uniformly continuous 5510 Lipschitz Primer page 4 Theorem 4
locally Lipschitz alone does not imply differentiable 5510 Lipschitz Primer page 5 Example 4
locally Lipschitz alone does not imply Lipschitz 5510 Lipschitz Primer page 4 Example 3
locally Lipschitz implies continuous 5510 Lipschitz Primer page 4 Theorem 4
locally Lipschitz on a set definition 5510 Lipschitz Primer page 2 Definition
locally Lipschitz on compact set implies Lipschitz 5510 Lipschitz Primer page 3 Theorem 3
lp spaces 5510 II.1 page 4 (Example)

M
maximal ideal definition 5410 III.2 page 9 Definition III.2.17
maximal ideals always exist in rings with identity 5410 III.2 page 9 Theorem III.2.18
maximal left ideal defintion 5410 III.2 page 9 Definition III.2.17
maximal normal subgroup of G definition 5410 II.8 page 2 Definition.
maximal right ideal definition 5410 III.2 page 9 Definition III.2.17
meaningful product in a semigroup 5410 I.1 page 9
Meromorphic Function Definition 5510 Extended Complex Plane page 5 Note
metacyclic group definition 5410 II.6 page 1 Proposition II.6.1 (ii)
Metric Space Definiton 5510 II.1 page 1 (Definition)
Metric Space properties 5510 II.1 page 6 (Theorem II.1.9, Theorem II.1.11)
metric space properties 5510 II.4 page 5 (Proposition II.4.9)
Metric Theorem for ℂ∞ 5510 Extended Complex Plane page 1 Metric Theorem ℂ∞
minimal normal subgroup of a group G definition 5410 II.7 page 8 Definition.
minimal normal subgroup of a solvable group is an abelian p-group 5410 II.7 page 9 Lemma II.7.13
Modulus of Complex Number 5510 I.2 page 2 (Definition)
monoid 5410 I.1 page 2 (Definition I.1.1)
Monomorphism 5410 I.2 page 1 (Definition I.2.1)
Monomorphism, Characterization 5410 I.2 page 3 (Theorem I.2.3)
monomorphism of quotient ring to product of quotient rings 5410 III.2 page 14 Corollary III.2.27
monomorphism of rings definition 5410 III.1 page 7 Note
morphism definition 5410 I.7 page 2 Definition I.7.1 (i)
Multiplication of Complex Numbers 5510 I.2 page 1 (Definition)
Multiplicative Inverse of complex number 5510 I.2 page 2 (Notice)

N
nilpotent definition 5410 II.3 page 5 Definition.
nilpotent definition 5410 II.7 page 2 Definition II.7.1
nilpotent finite group characterization 5410 II.7 page 3 Proposition II.7.5
no group of order 15 is simple 5410 II.5 page 8 Example. Fraleigh, Example 36.13
nonabelian groups of order 12 5410 II.6 page 2 Proposition II.6.4
nonabelian groups of order 8 5410 II.6 page 2 Proposition II.6.3
nonempty subset of a ring is a left or right ideal characterization 5410 III.2 page 2 Theorem II.2.2
Norm 5510 II.1 page 3 (Example)
normal endomorphism definition 5410 II.3 page 5 Definiition
normality of intersections and joins of subgroups 5410 I.5 page 2 (Theorem I.5.3)
normalizer of normalizer of a sylow p-subgroup is the normalizer of the sylow p-subgroup 5410 II.5 page 6 Theorem II.5.11
normalizer of subgroup K in group G definition 5410 II.4 page 3 Definition.
normal series of a group G definition 5410 II.8 page 1 Definition II.8.1
normal subgroup definition 5410 I.5 page 1 (Definition I.5.2)
number of elements in conjugacy class of group element x is index of stabilizer of x5410 II.4 page 4 Corollary II.4.

O
odd permutation definition 5410 I.6 page 3 Definition I.6.6
One-point compactification definition 5510 Extended Complex Plane page 4 Definition
one-step refinement of a subnormal series definition 5410 II.8 page 2 Definition II.8.2
Only One Complete Ordered Field 5510 I.1 page 1 (Theorem)
only one sylow p-subgroup P implies P normal 5410 II.5 page 5 Corollary II.5.8 (iii)
Open Ball 5510 II.1 page 1 (Definition)
open cover defintition 5510 II.4 page 1 (Definition II.4.1)
Open definition 5510 II.1 page 5 (Definition II.1.8)
only divisors of irreducible ring element are its associates and its units 5410 III.3 page 3 Theorem III.3.4
orbit of x definition 5410 II.4 page 3 Definition.
Ordering 5510 Ordering-ℂ page 3 (Note)
Ordering of a Field 5510 I.1 page 1 (Recall)
order of a permutation definition 5410 I.6 page 3 Corollary I.6.4
order of group element 5410 I.3 page 1 (Definition I.3.3)
order of product of subgroups 5410 I.4 page 3 (Theorem I.4.7)

P
parity of a permuation 5410 I.6 page 3 Definition I.6.6
partial sum of a series definition 5510 III.1 page 1 Definition
permutation can only be even or odd 5410 I.6 page 3 Theorem I.6.7
p-group charcterization 5410 II.5 page 4 Corollary II.5.3
p-group definition 5410 II.5 page 4 Definition.
Pogorui and Shaprio's theorem holds for analytic functions of a quaternional polynomial 5410 Quaternions Algebraic Supplement page 21 Note.
Pole of a Function Definition 5510 Extended Complex Plane page 5 Note
Pole of complex-valued function 5510 I.6 page 1 (Note)
polygon from a to b definition 5510 II.2 page 2 (Definition)
polynomials can be factored into a product of linear terms in left or right algebraically closed division rings 5410 Quaternions Algebraic Supplement page 18 Note.
polynomials of degree n have at most n distinct roots in an integral domain which contains it 5410 Quaternions Algebraic Supplement page 9 Hungerford's Theorem III.6.7
power series about a point definition 5510 III.1 page 3 Definition
presentation of a group definition 5410 I.9 page 7 Definition I.9.4
prime elements are irreducible 5410 III.3 page 2 Theorem III.3.4
prime ideal definition 5410 III.2 page 8 Definition III.2.14
prime ring element definition 5410 III.3 page 2 Definition III.3.3
principal ideal definition 5410 III.2 page 3 Definition III.2.4
principal ideal domain definition 5410 III.2 page 3 Definition III.2.4
principal ideal domains are unique factorization domains 5410 III.3 page 5 Theorem III.3.7
principal ideal ring definition 5410 III.2 page 3 Defintion III.2.4
prodcuts of series 5510 III.1 page 4 Proposition III.1.6
product definition 5410 I.7 page 6 Definition I.7.2
product of subsets of a group 5410 I.4 page 3 (Note)
product of two nonempty subsets of a ring definition 5410 III.2 page 4 Definition
product of two quaternionic polynomials defintion 5410 Quaternions Algebraic Supplement page 11 Definition.
proper ideal definition 5410 III.2 page 2 Note.
proper refinment of a subnormal series definition 5410 II.8 page 2 Definition II.8.2
Proper Subgroup 5410 I.2 page 4 (Definition I.2.4)
properties of compact sets 5510 II.4 page 3 (Proposition II.4.3)
properties of normal subgroups 5410 I.5 page 1 (Theorem I.5.1)
Prüfer group (group of rationals modulo 1) 5410 I.1 page 8
p-subgroup of G definition 5410 II.5 page 4 Definition.

Q
quaternionic conjugate definition 5410 Quaternions Algebraic Supplement page 16 Definition.
quaternionic modulus definition 5410 Quaternions Algebraic Supplement page 16 Note.
quaternionic polynomial definition 5410 Quaternions Algebraic Supplement page 10 Note.
quaternions are left and right algebraically closed 5410 Quaternions Algebraic Supplement page 18 Theorem 16.14 ("Niven-Jacobson" in Lam) Fundamental Theorem of ALgebra for Quaternions
quaternions form a noncommutative division ring 5410 Quaternions Algebraic Supplement page 3 Theorem
quotient group has a ring structure 5410 III.2 page 4 Note.
quotient group is abelian if and only if subgroup contains the commutator subgroup 5410 II.7 page 4 Theorem II.7.8
quotient group isomorphism not guaranteed 5410 I.5 page 10 (Note)
quotient group of a commutative ring is commutative 5410 III.2 page 5 Theorem III.2.7
quotient group of a ring with idenity has identity 5410 III.2 page 5 Theorem III.2.7
quotient group of direct products isomoprhic to direct product of the quotient groups 5410 I.8 page 9 Corollary I.8.11
quotient group of external weak direct products is isomorphic to external weak direct products of quotient groups 5410 I.8 page 9 Corollary I.8.11
quotient group (see also factor group) 5410 I.5 page 3 (Theorem I.5.4, Definition, Note)
quotient ring is a division ring implies ideal is maximal in rings with identity 5410 III.2 page 10 Theorem III.2.20
quotient ring is a field where the ring has identity and the ideal is maximal 5410 III.2 page 10 Theorem III.2.20

R
radius of convergence definition 5510 III.1 page 3 Theorem III.1.3
rank of free abelian group definition 5410 II.1 page 3 Definition
ratio test for complex power series 5510 III.1 page 4 Proposition III.1.4
Real Numbers 5510 I.1 page 1 (Definition)
real quaternions division ring definition (see also division ring of real quaternions) 5410 III.1 page 5 Example
real quaternions division ring may be interpreted as a subring of 2x2 matrices over C 5410 III.1 page 6 Example
Recognition Theorem for Direct Products 5410 Semidirect Product Supplement page 3 Theorem DF.5.9
Recognition Theorem for Semidirect Products 5410 Semidirect Product Supplement page 12 Theorem DF.5.12
reduced word definition 5410 I.9 page 2 Definition
reduced word in product of groups definition 5410 I.9 page 11 Definition
refinement of a subnormal series definition 5410 II.8 page 2 Definition II.8.2
refinements of composition series are equivalent to the original composition series5410 II.8 page 4 Lemma II.8.8
relation on generators of group G 5410 I.9 page 6 Note 1
Relationship between modulus and conjugate 5510 I.2 page 2 (Notice)
relatively prime ring elements definition 5410 III.3 page 6 Definition III.3.10
right and left congruence modulo H are equivalenace relations on G 5410 I.4 page 1 (Theorem I.4.2)
right and left congruence modulo subgroup H 5410 I.4 page 1 (Definition I.4.1)
right and left coset definition 5410 I.4 page 1 (Theorem I.4.2)
right and left coset properties 5410 I.4 page 2 (Corollary I.4.3)
right and left cosets partition G 5410 I.4 page 2 (Note)
right ideal generated by a set X definition 5410 III.2 page 3 Definition III.2.4
right ideal of a ring R definition 5410 Quaternions Algebraic Supplement page 14 Note.
right ideal subring definition 5410 III.2 page 1 Definition III.2.1
right inverse definition 5410 III.1 page 3 Defintiion III.1.4
right invertible ring element definition 5410 III.1 page 3 Definition III.1.4
right root definition 5410 Quaternions Algebraic Supplement page 13 Definition 16.1 of Lam.
right zero divisor definition 5410 III.1 page 2 Definition III.1.3
ring definition 5410 III.1 page 1 Definition III.1.1
ring identities/properties 5410 III.1 page 2 Theorem III.1.2
ring of nxn matrices over a division ring has no propoer two sided ideals 5410 III.2 page 2 Example.
ring with identity definition 5410 III.1 page 1 Definition III.1.1
ring with no zero divsors has prime characteristic 5410 III.1 page 7 Theorem III.1.9 (iii)
ring with unity definition 5410 III.1 page 1 Definition III.1.1
r is a left root if and only if t-r is a left divisor 5410 Quaternions Algebraic Supplement page 13 Proposition 16.2 of Lam. (The Factor Theorem in a Ring with Unity).
r is a right root if and only if t-r is a right divisor 5410 Quaternions Algebraic Supplement page 13 Proposition 16.2 of Lam. (The Factor Theorem in a Ring with Unity).
roots of polynomials of degree n in a division ring lie in at most n conjugacy classes 5410 Quaternions Algebraic Supplement page 15. Theorem 16.4 of Lam. ("Gordon-Motzkin" in Lam.)
roots of products of polynomials in a division ring 5410 Quaternions Algebraic Supplement page 14 Proposition 16.3 of Lam.
roots of quaternionic polynomials are circles 5410 Quaternions Algebraic Supplement page 11 Theorem.

S
Schrier's Theorem 5410 II.8 page 6 Theorem II.8.10
Second Isomorphism Theorem 5410 I.5 page 6 (Corollary I.5.9)
Second Isomorphism Theorem 5410 III.2 page 7 Theorem III.2.12 (i)
second sylow theorem 5410 II.5 page 6 Theorem II.5.9
semidirect product of two groups with respect to a homomorphism definition 5410 Semidirect Product Supplement page 6-7 Theorem DF.5.10
semigroup 5410 I.1 page 2 (Definition I.1.1)
Separable space definition 5510 II.1 page 9 (Note)
separation of a set 5510 II.2 page 2 (Note)
sequentially compact definition 5510 II.4 page 4 (Definition II.4.7)
set of all units in a ring with identity form a multiplicative group 5410 III.1 page 3 Note
set of roots of a polynomail in H[q] consists of isolated points or isolated two dimensional spheres 5410 Quaternions Algebraic Supplement page 20 Theorem (Poguri and Shapiro).
sign of a permutation defintion 5410 I.6 page 3 Definition I.6.6
simple group definition 5410 I.6 page 7 Definition I.6.9
S_n is not solvable for n >= 5 5410 II.7 page 7 Corollary II.7.12
solvable group defined in terms of composition series 5410 II.7 page 6 Definition 35.18
solvable group definition 5410 II.7 page 6 Definition II.7.9
solvable groups of order a product of two relatively prime numbers properties 5410 II.7 page 9 Proposition II.7.14
solvable if and only if has a solvable series 5410 II.8 page 3 Theorem II.8.5
solvable series definition 5410 II.8 page 2 Definition II.8.3
stabilizer of x definition 5410 II.4 page 2 Theorem II.4.2 ii
standard n product in a semigroup 5410 I.1 page 10
Stereographic projections in ℂ 5510 I.6 page 2 (Definition)
subgoup fixing x definition 5410 II.4 page 2 Theorem II.4.2 ii
Subgroup 〈X〉 Generated by X Characterization 5410 I.2 page 6 (Theorem I.2.8)
Subgroup 5410 I.2 page 4 (Definition I.2.4)
Subgroup, Characterization of 5410 I.2 page 5 (Theorem I.2.5)
subgroup of a finitely generated abelian group is finitely generated 5410 II.1 page 6 Corollary II.1.7
subgroup of a free abelian group is free abelian 5410 II.1 page 5 Theorem II.1.6
Subgroup of G Generated by the Set X 5410 I.2 page 5 (Definition I.2.7)
subgroups of additive group Z are cyclic 5410 I.3 page 1 (Theorem I.3.1)
subgroups of solvable groups are solvable 5410 II.7 page 7 Theorem II.7.11 (i)
subgroup with smallest prime index is normal 5410 II.4 page 6 Corollary II.4.10
subnormal series is a composition series if and only if it has no proper refinements 5410 II.8 page 3 Theorem II.8.4
subnormal series of a group G definition 5410 II.8 page 1 Definition II.8.1
subring definition 5410 III.2 page 1 Definition III.2.1
sufficient conditions for finite groups of order a product of two relatively prie numbers to be solvable 5410 II.7 page 10 Theorem.
sums of series 5510 III.1 page 4 Proposition III.1.6
sylow p-subgroup definition 5410 II.5 page 5 Definition.
symmetric group 5410 I.1 page 6

T
table of groups of a particular order 5410 II.6 page 4
Taxicab Metric 5510 II.1 page 2 (Example)
terminal object definition (see also couniversal object) 5410 I.7 page 10 Definition I.7.9
Third Isomorphism Theorem 5410 III.2 page 7 Theorem III.2.12 (ii)
Third Isomorphism Theroem 5410 I.5 page 8 (Corollary I.5.10)
third sylow theorem 5410 II.5 page 6 Theorem II.5.10
topolgically complete definition 5510 II.4 page 8 (Note)
Topological Space definition 5510 II.1 page 7 (Definition)
Topologies on ℂ∞ 5510 Extended Complex Plane page 3 Topologies on ℂ∞ Theorem
torsion free group definition 5410 II.2 page 3 Definition
torsion group definition 5410 II.2 page 3 Definition
torsion subgroup definition 5410 II.2 page 3 Definition
total boundedness definition 5510 II.4 page 5 (Proposition II.4.9)
Total Ordering 5510 Ordering-ℂ page 3 (Note)
transposition definition 5410 I.6 page 1 Definition I.6.1
Triangle Inequality 5510 I.3 page 2 (Theorem I.3.A)
Trichotomy Property 5510 Ordering-ℂ page 1 (3)
Trivial Subgroup 5410 I.2 page 4 (Definition I.2.4)
trivial subgroup is a p-subgroup 5410 II.5 page 4 Note.
two free objects on the same set are equivalent 5410 I.7 page 10 Theorem I.7.8
type of a permutation definition 5410 I.6 page 2 Note.
types of subgroups of abelian groups which are guaranteed to exist 5410 II.2 page 2 lemma II.2.5

U
uniform continuity alone does not imply Lipschitz 5510 Lipschitz Primer page 4 Example 2
uniform convergence definition 5510 II.6 page 1 Definition
uniformly continuous definition 5510 II.5 page 4 Definition II.5.6
uniformly continuous does not imply Lipschitz 5510 II.5 page 4 Example
union of connected sets with common point is connected 5510 II.2 page (Lemma II.2.6)
unique factorization domain definition 5410 III.3 page 3 Definition III.3.5
unit definition 5410 III.1 page 3 Definition III.1.4
universal object definition (see also inital object) 5410 I.7 page 10 Definition I.7.9

V
von Dyck's Theorem 5410 I.9 page 8 Theorem I.9.5

W
Weierstrass M-test 5510 II.6 page 2 Theorem II.6.2
Well-Ordering Principle 5510 Ordering-ℂ page 3 (Note)
word definition 5410 I.9 page 1 Defintion

X

Y

Z
Zassenhaus' Lemma 5410 II.8 page 5 Lemma II.8.9
zero divisor definition 5410 III.1 page 2 Definition III.1.3
14 groups of order 16 5410 II.6 page 5 Note.

Return to Bob Gardner's home page

Sections included:
Modern Algebra (5410) I.1, I.2, I.3, I.4, I.5, I.6, I.7, I.8, I.9, II.1, II.2, II.3, II.4, II.5, II.6, Semidirect Product, II.7, II.8, III.1, Quaternion Supplement, III.2, III.3, III.4
Complex Analysis (5510) I.1, I.2, I.3, I.4, I.5, I.6, Ilieff-Sendov, Ordering-ℂ, II.1, II.2, II.3, II.4, Extended-Complex-Plane, II.5, II.6, Lipschitz Primer, III.1

Prepared with the assistance of Nicholas Carney, Fall 2017, Spring 2018.
Bob Gardner Index of Class Notes/revised April 22, 2018.