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

HW3, due Wed Feb 1, 2017

In class on January 25, we went over a large number of exercises.  You will not be asked to submit solutions on February 1.  Instead, read all of the exercises from Sections 2.1 (17 problems), 2.2 (14 problems), and 2.3 (26 problems).

 

Submit a sheet of paper stating for each problem which category each problem falls into:

  • (Solved) You have solved the problem.
  • (Understand)  You have not solved the problem, but you are convinced that you understand exactly how to solve the problem.
  • (Unsolved) You are not convinced you understand how to solve the problem.

 

HW2, due Wed Jan 25, 2017

  • DF Sec 1.3 #5, #8, #14
  • DF Sec 1.6 #9, #17, #23,
  • DF Sec 1.7 #17
  • (wording updated on Jan 23.) Let F be a field.  Let V=F^n, viewed as an n-dimensional vector space over the field F.  Show the following:
    • The group G=GL(n,F) acts on V by linear transformations.
    • If B=(e_1,\ldots,e_n) is  the standard ordered basis of V and if g \in G, then g B = (g e_1,\ldots,g e_n)  is an ordered basis of V.
    • There is a 1-1 correspondence between elements of GL(n,F) and bases of V.