Math 301 §A: Objectives for Second Midterm Exam

State the Definitions and use them in Proofs.

• Cyclic group
• Order of a group; order of a group element
• Permutation; Permutation Group
  • Symmetric group on a set
  • Cycle; transposition
  • Inversion; even permutation; odd permutation
  • Alternating group
  • Stabilizer of a point
  • Orbit of a point
• Isomorphism of groups
  • Automorphism of a group
  • Inner automorphism of a group
• (Left) Cosets of a subgroup
• Congruence mod a subgroup H of a group G:
       a ∼ b mod H if a^-1b ∈ H.
• [G:H] -- the index of subgroup H in group G.

State the Theorems and use them in Proofs.

• Fundamental Theorem of Cyclic Groups
• Number of elements of given order in a cyclic group
• Permutations of finite sets:
  • Every permutation can be written as a product of disjoint cycles.
  • Disjoint cycles commute.
  • Every permutation can be written as a product of transpositions.
  • Order of a permutation is LCM of lengths of its cycles.
  • Even permutations in S_n form a subgroup A_n of order n! / 2.
• Cayley's Theorem;
  The left regular representation of a group
• Properties of group elements preserved by isomorphisms
• Properties of groups preserved by isomorphisms
• If G is a group, Aut(G) and Inn(G) are groups.
• Lagrange's Theorem, and corollaries:
  • The order of an element divides the order of the group.
  • The order of a subgroup divides the order of the group.
  • Every group of prime order is cyclic.
  • Fermat's Little Theorem.
• Orbit-Stabilizer Theorem.

Computations.

• Display the subgroup lattice of a cyclic group.
• Write a permutation in S_n as
  • A product of disjoint cycles
  • A product of transpositions

Recognize, define, and work with common Groups.

• Aut(Z_n) = U(n).
• Rotation groups of
  • Regular Tetrahedron (A₄)
  • Cube (S₄) and
  • Soccer Ball (A₅).

alex at iastate dot edu

Document last modified on Mon Feb 28 2011