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()

 

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