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));

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