Math 301 §A: Objectives for First Midterm Exam

State the Definitions and use them in Proofs.

• Greatest Common Divisor in Z.
• Equivalence Relation; Partition.
• Function;
  • One-to-one function;
  • Onto function;
  • Composition of functions.
• Group
  • Abelian (Commutative) group; non-Abelian group.
  • Order of a group; order of a group element.
  • Subgroup
  • Cyclic subgroup ⟨a⟩ generated by a.
  • Center of a group.
  • Centralizer of a group element.

State the Theorems and use them in Proofs.

• Division algorithm in Z.
• The GCD is a linear combination.
• Euclid's Lemma.
• Fundamental Theorem of Arithmetic.
• Well-ordering is equivalent to Induction.
• Equivalence relations and partitions:
  • If R is an equivalence relation on a set S,
    the equivalence classes under R form a partition of S.
  • Conversely, given a partition of a set S, there exists
    an equivalence relation on S whose equivalence classes
    are the subsets in the partition.
• Composition of functions is associative.
• A one-to-one and onto function has an inverse function.
• DeMoivre's Theorem.
• Theorems about Groups:
  • Uniqueness of identity and inverses;
  • Cancellation laws;
  • Inverse of a product;
  • Subgroup tests;
  • The Center of a Group is a Subgroup;
  • The Centralizer of an Element is a Subgroup.

Computations and Proof Techniques.

• Use Euclid's algorithm to find GCD(a,b)
  and to find r,s so GCD(a,b)=ra+sb.
• Do modular arithmetic in Z_m.
• Construct proofs-by-induction.

Recognize, define, and work with common Groups.

• Dihedral groups D_n.
• Subgroups of (R,+) and (R*, ×).
• Groups using modular arithmetic (Z_m,+) and (U(m),×).
• The group of complex n^th roots of unity {cos(2πk/n) + i sin(2πk/n): 0≤k≤n-1}.
• Matrix Groups
  • General linear groups GL(n,F) of invertible
    n×n matrices over a field F = Q, R, C or Z_p.
  • Special linear groups SL(n,F) of n×n matrices
    having determinant 1, over ring F = Z or a
    field F = Q, R, C or Z_p.
• Translation groups (Fⁿ,+), for F = Z, Q, R, C.

alex at iastate dot edu

Document last modified on Sun Feb 6 2011