Math 5310
Intr. to Modern Algebra I
MWF 12:55-1:45pm
Room: LCB 121
Fall 2004
- Test # 1 - Monday 9/20
- Here is a list of Problems for you to do to
prepare for Test # 1
- Here is a list of definitions and theorems that you are expected to know
when you take Test # 1:
-
Section 1.2: Set Theory
- Definition of intersection of two sets (page 4)
- Definition of union of two sets (page 4)
- Definition of difference of two sets (page 4)
-
Section 1.3: Mappings
- Definition of function (page 8)
- Definition of onto function (page 10)
- Definition of 1-1 function (page 10)
- Definition of image of a set under a function (page 10)
- Definition of inverse image of a set under a function (page 10)
- Definition of composition of functions (page 11)
-
Section 1.4: A(S)
- Definition of A(S) (page 16)
-
Section 1.5: The integers
- Theorem 1.5.1 (Euclid's Algorithm, page 22)
- Definition of an integer dividing another integer (page 22)
- Definition of greatest common divisor (page 23)
- Theorem 1.5.3 (page 23)
- Definition of two integers being relatively prime (page 25)
- Theorem 1.5.4 (page 25)
-
Section 1.6: Mathematical Induction
-
Section 1.7: Complex Number
- Definition of a complex number (page 32)
- Definition of real part of a complex number (page 32)
- Definition of imaginary part of a complex number (page 32)
- Definition of complex conjugate of a complex number (page 32)
- Formula for the inverse of a complex number (page 33)
- Definition of absolute value of a complex number (page 34)
- Definition of polar form of a complex number (page 35)
- Theorem 1.7.6 (De Moivre's Theorem, page 36)
- Test # 2 - Monday 10/25
- Here is a list of definitions and theorems that you are expected to know
when you take Test # 2:
- Section 2.1: Definitions and Examples of Groups
- Section 2.3: Subgroups
- Definition of subgroup.
- Definition of the cyclic subgroup generated by an element a of G.
- Definition of cyclic group.
- Definition of the center of a group.
- Definition of the centralizer of an element a of G.
- Section 2.4: Lagrange's Theorem
- Theorem 2.4.2 (Lagrange's Theorem).
- Definition of index of H in G.
- Definition of order of an element a of G.
- Theorem 2.4.4.
- Theorem 2.4.5.
- Section 2.5: Homomorphisms and Normal Subgroups
- Definition of homomorphism.
- Definition of monomorphism.
- Lemma 2.5.2.
- Lemma 2.5.3.
- Definition of kernel of a homomorphism.
- Theorem 2.5.5.
- Definition of normal subgroup.
- Theorem 2.5.6.
- Here is a list of Problems for you to do to prepare for Test # 2:
- Section 2.3: Problems 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 26, 29.
- Section 2.4: Problems 6, 7, 8, 9, 12, 13, 14, 16, 17, 27, 30.
- Section 2.5: Problems 3, 4, 5, 7, 12, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 36, 38, 40, 41, 42, 43, 44.
- Test # 3 - Monday 11/22
- Here is a list of definitions and theorems that you are expected to know
when you take Test # 3:
- Section 2.6: Factor Groups
- Definition of G/N.
- Theorem 2.6.1.
- Theorem 2.6.2.
- Theorem 2.6.3.
- Section 2.7: The Homomorphisms Theorems
- Theorem 2.7.1 (First Homomorphism Theorem).
- Theorem 2.7.2 (Correspondence Theorem).
- Theorem 2.7.3 (Second Homomorphism Theorem).
- Theorem 2.7.4 (Third Homomorphism Theorem).
- Section 2.8: Cauchy's Theorem
- Theorem 2.8.2 (Cauchy's Theorem).
- Section 2.10: Finite Abelian Groups
- The Classification of Finite Abelian Groups.
- Section 2.11: Conjugacy and Sylow's Theorem
- Theorem 2.11.2.
- Theorem 2.11.3 (The Class Equation).
- Theorem 2.11.4.
- Theorem 2.11.5.
- Here is a list of Problems for you to do to prepare for Test # 3:
- Section 2.6: Problems 1, 2, 3, 4, 5, 7, 8, 11, 12, 13, 15, 17.
- Section 2.7: Problems 2, 3, 4, 5, 6, 7.
- Section 2.8: Problems 2, 7, 8, 9, 10, 11.
- Section 2.9: Problems 2, 5, 6.
- Section 2.10: Problems 1, 2, 3.
- Final - Tuesday 12/14 - 1:00-3:00pm
- Here is a list of the definitions that I might ask you to write on the
Final:
- Section 2.3: Subgroups
- Definition of subgroup.
- Definition of the cyclic subgroup generated by an element a of G.
- Definition of cyclic group.
- Definition of the center of a group.
- Definition of the centralizer of an element a of G.
- Section 2.4: Lagrange's Theorem
- Definition of index of H in G.
- Definition of order of an element a of G.
- Section 2.5: Homomorphisms and Normal Subgroups
- Definition of homomorphism.
- Definition of monomorphism.
- Definition of kernel of a homomorphism.
- Definition of normal subgroup.
- Section 2.6: Factor Groups
- Section 2.9: Direct Products
- Definition of direct product of groups.
- Here is a list of the theorem that you can use on the Final without having
to prove them, with the exception of the ones marked with *. For those, you
need to be able to write the proof if there is a question asking just that.
- Section 2.4: Lagrange's Theorem
- Theorem 2.4.2 (Lagrange's Theorem).
- Theorem 2.4.4.
- Theorem 2.4.5.
- Section 2.5: Homomorphisms and Normal Subgroups
- Lemma 2.5.2.
- Lemma 2.5.3.
- * Theorem 2.5.5.
- * Theorem 2.5.6.
- Section 2.6: Factor Groups
- Theorem 2.6.1.
- Theorem 2.6.2.
- Theorem 2.6.3.
- Section 2.7: The Homomorphisms Theorems
- Theorem 2.7.1 (First Homomorphism Theorem).
- Theorem 2.7.2 (Correspondence Theorem).
- Theorem 2.7.3 (Second Homomorphism Theorem).
- Theorem 2.7.4 (Third Homomorphism Theorem).
- Section 2.8: Cauchy's Theorem
- Theorem 2.8.2 (Cauchy's Theorem).
- * Lemma 2.8.3 (also with q replaced by any number n, not necessarely
prime).
- Section 2.9: Direct Products
- Lemma 2.9.2.
- * If M and N are two normal subgroups of G such that the intersection of M
and N is (e), then there exists a 1-1 homomorphism from M x N to G.
- * Let G be an abelian group. If A and B are two subgroups of G, then
there exists a homomorphism from A x B to G with kernel isomorphic to the
intersection of A and B.
- Section 2.10: Finite Abelian Groups
- The Classification of Finite Abelian Groups.
- Here is a list of Problems for you to do to prepare for the Final:
- Section 2.3: Problems 8, 9, 10, 13, 16, 19, 20, 29.
- Section 2.4: Problems 6, 7, 9, 12, 16, 17.
- Section 2.5: Problems 12, 14, 15, 17, 19, 26.
- Section 2.6: Problems 1, 2, 3, 4, 5, 7, 8, 17.
- Section 2.7: Problems 2, 3, 5.
- Section 2.8: Problems 7, 8, 9, 10.
- Section 2.9: Problems 2, 5, 6 (Do Problems 2.9.5 and 2.9.6 only for the
case of TWO subgroups N_1 and N_2).
- Section 2.10: Problems 1, 2, 3.
Last updated on December 15, 2004.