HW9, due Wed Mar 22

  • Let F(A) and F'(A) both satisfy the universal property of a free group on a set A.  Show that there is a unique isomorphism from F(A) to F'(A) that is compatible with the canonical maps A -> F(A) and A -> F'(A).  Hint: consider the universal property for all possible combinations, taking both F(A) and F'(A) to be the free group and the supplementary group.
  • Can you see how a similar argument could be used to show quite generally that objects that satisfy universal properties are unique up to isomorphism?
  • DF 5.4 #2,
  • DF 5.4 #7,
  • DF 5.5 #23-a

HW8, due Friday March 17

Do the following calculations in Sage (not GAP). You may use Sage Cell for the calculations and cut and paste the answers that you submit.  You are welcome to improve the code.

 

  1. compute 77+31 (no semicolon!)
  2. compute the set of divisors of 60:  divisors(60)
  3. compute the greatest common divisor: gcd(33,770)
  4. compute 17 mod 3: mod(17,3)
  5. compute a and b such that a*64+b*33 = 1, using xgcd(64,33) and check your answer in Sage
  6. find factor(2^126-1)
  7. check the largest factor is prime: is_prime(77158673929)
  8. compute the square of the permutation
    • G = SymmetricGroup(7)
    • a = G(“(1,2,3,4)(6,5,7)”)
    • a^2
  9. compute the “left to right” (warning!) product of the cycles
    • G = SymmetricGroup(3)
    • a = G(“(1,2,3)”)
    • b= G(“(2,3)”)
    • a*b
  10. make a list of elements of the permutation group generated by (1,2,3,4,5) and (2,5)(3,4):
    • G = SymmetricGroup(5)
    • g1 = G(“(1,2,3,4,5)”)
    • g2 =G(“(2,5)(3,4)”)
    • H = G.subgroup([g1,g2])
    • H.list()
  11. Calculate the conjugacy classes of A5:
    • G =  AlternatingGroup(5)
    • G.conjugacy_classes_representatives()
  12. Find the derived subgroup of A4:
    • G=AlternatingGroup(4)
    • G.commutator()
  13. Calculate a 2-Sylow subgroup of S4: SymmetricGroup(4).sylow_subgroup(2)
  14. Show that the subgroup of S4 generated by (1,2,3) and (2,3,4) is A4:
    •  G=SymmetricGroup(4)
    • g1=G(“(1,2,3)”)
    • g2=G(“(2,3,4)”)
    • H=G.subgroup([g1,g2])
    • H.is_isomorphic(AlternatingGroup(4))
  15. Calculate a composition series for S4: SymmetricGroup(4).composition_series()