Exam Feb 22, 2017

The exam will cover up to and including Section 4.2.

 

The bonus problems (not to be presented during the exam) are as follows:

 

  • Given the category of finite abelian groups, can you determine which object is which group (up to isomorphism)?
  • Determine the group generated by two adjacent faces of the Rubik’s cube.

HW 4, due Wed Feb 8, 2017

  • DF Section 2.4 #9 #17 (Zorn’s lemma appears in Appendix 1), #18
  • DF Section 3.1 #1, #12.
  • Let F be any field.  Show that there is a 1-1 homormophism from S_n to $GL(n,F)$ given by the action of S_n on the indices i in the standard basis e_1,\ldots,e_n.  Combine this with Cayley’s theorem (page 120) to show that for every finite group G, there exists n, such that G is isomorphic to a subgroup of GL(n,F).
  • Show algebraically that every isometry T of R^2 is a glide reflection provided it has the form $\latex T=T_\lambda A$, where $A\in O(2,R)$, $A\not\in SO(2,R)$, $\latex \lambda\ne 0$.
  • Solve the following problems without submitting a solution
    • DF Section 2.4 #15, #19