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.

- compute 77+31;
- compute the set of divisors of 60: DivisorsInt(60);
- compute the greatest common divisor Gcd(33,770);
- compute 17 mod 3;
- compute a and b such that a*64+b*33 = 1, using GcdRepresentation(64,33); check your answer in gap
- find Factors(2^126-1);
- check the largest factor is prime: IsPrime(77158673929);
- compute the square (a^2;) of the permutation
*a:=(1,2,3,4)(6,5,7);*
- compute the
**“left to right” (warning!) **product of the cycles *a:= (1,2,3); b:=(2,3); a*b;*
- 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);*
- Calculate the conjugacy classes of A5:
*g:= AlternatingGroup(5);*

* c:= ConjugacyClasses(g);*
- Find the derived subgroup of A4:
*DerivedSubgroup(AlternatingGroup(4));*
- Calculate the 2-Sylow subgroup of S4:
*SylowSubgroup(SymmetricGroup(4),2);*
- 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)));*
- Calculate a composition series for S4:
*DisplayCompositionSeries(SymmetricGroup(4));*

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