Math 5310
Intr. to Modern Algebra I

• 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
      • Theorem 1.6.1 (page 29)
    • 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
      • Definition of group.
    • 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
      • Definition of G/N.
    • 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.