Schedule, Monday March 20-Friday March 31

  • Monday 3/20, Hales, review of rings, ideals, fields, and extensions
  • Wednesday 3/22, Fedorov, D&F Section 13.1, basic theory of field extensions
  • Friday 3/24, Fedorov, D&F Section 13.2, algebraic extensions
  • Monday 3/27, Xu, D&F, Section 13.3, classical straightedge and compass constructions
  • Wednesday 3/29, Yarosh, D&F, Section 13.4 splitting fields
  • Friday 3/29, Yarosh, D&F, Section 13.4, algebraic closures
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HW 7, due Wednesday March 25, 2017

Perform the following calculations in GAP.  You can use Sage Cell and choose GAP in the language menu at the bottom-right-hand corner of the input cell.  You can cut and paste your answers into the submitted homework.

  1.  compute 77+31;
  2. compute the set of divisors of 60:  DivisorsInt(60);
  3. compute the greatest common divisor Gcd(33,770);
  4. compute 17 mod 3;
  5. compute a and b such that a*64+b*33 = 1, using GcdRepresentation(64,33); check your answer in gap
  6. find Factors(2^126-1);
  7. check the largest factor is prime: IsPrime(77158673929);
  8. compute the square (a^2;) of the permutation a:=(1,2,3,4)(6,5,7);
  9. compute the “left to right” (warning!) product of the cycles a:= (1,2,3); b:=(2,3); a*b;
  10. make a list of representatives of the permutation group generated by (1,2,3,4,5) and (2,5)(3,4): g:= Group((1,2,3,4,5),(2,5)(3,4)); c:= ConjugacyClasses(g);
  11. Calculate the conjugacy classes of A5: g:= AlternatingGroup(5);
    c:= ConjugacyClasses(g);
  12. Find the derived subgroup of A4: DerivedSubgroup(AlternatingGroup(4));
  13. Calculate the 2-Sylow subgroup of S4: SylowSubgroup(SymmetricGroup(4),2);
  14. Show that the subgroup of S4 generated by (1,2,3) and (2,3,4) is A4: StructureDescription(Group((1,2,3),(2,3,4)));
  15. Calculate a composition series for S4: DisplayCompositionSeries(SymmetricGroup(4));

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.