GAP Lesson 1

Start up GAP.

gap> LogTo("GAPlesson1");

Arithmetic Operations

You learn: + - * / ^, terminating input with ;, last. Factors, mod, Int, Gcd, Lcm, Factorial.

Try the following. Press the return key at the end of each line.

gap> 5+7;
gap> 12/17;
gap> 2/3 + 3/4;
gap> 2^3*3^3;
gap> 3.14159;

Discuss.

gap> Factors(1111111);
gap> Factors(11111111111);
gap> Factors(2^32 + 1);
gap> Factors(216);
gap> last;
gap> Collected(last);

There are also variables last2 and last3. Guess what their values are.

Exercise: Find the first number of the form 111..1 which is prime. Note that such a number must have a prime number of digits. As well as Factors, another useful function is IsPrime.

gap> -5 mod 11;
gap> 6 mod -5;
gap> 6 mod 0;
gap> Int(295/7);
gap> Gcd(216,930);
gap> Lcm(216,930);
gap> Factorial(6);

What would you expect Int(-1/2) to be?

Permutations
You learn: permutations are enclosed in (), operation s have the same form as for integers. Conjugation is built in.

gap> (1,2,3)*(1,2);
gap> (1,2,3)^-1;
gap> (1,2,3)^(2,5);
gap> 1^(1,2,3);

When we come to them, other kinds of elements such as matrices, and elements of finite fields can be manipulated with the same syntax.

Help
You learn: lines are not terminated with ; How to call up and browse sections of the manual.

gap> ?
gap> ?Help
gap> ?Factor
gap> ?A f s
gap> ?>
gap> ??

True and False; Assignment of Variables
You learn: The difference between = and :=, how to store things in memory.

gap> 216=2^3*3^3;
gap> 216=2^3;
gap> g=17;
gap> g;
gap> g:=17;
gap> g;
gap> g=17;
gap> g=21;
gap> g^2;
gap> 3<=2;
gap> 3>=2;

Exercise: What response do you expect from
gap> true and false;
gap> true or false;
?

Permutation groups; groups from the library; operations on groups
You learn: how to input a group generated by permutations. The operations Size, Center, DerivedSubgroup, in, SylowSubgroup, Elements, SymmetricGroup, AlternatingGroup, DihedralGroup.

gap> a:=Group((2,3,5)(6,7,8),(1,2,4,7)(3,6,8,5));
gap> Size(a);
gap> Center(a);
gap> Size(last);
gap> g:=SymmetricGroup(6);
gap> Size(g);
gap> h:=DerivedSubgroup(g);
gap> Size(h);
gap> (1,2) in h;
gap> (1,2,3) in h;
gap> p:=SylowSubgroup(g,2);
gap> Size(p);
gap> Elements(p);
gap> Center(p);
gap> Size(last);
gap> d:=DihedralGroup(8);
gap> Size(d);

Exercise: Identify the Sylow 2-subgroups of
gap> c:=Group((1,2,3,4,5,6,7),(2,3)(4,7));
and
gap> b:=Group((2,3)(4,6)(5,7),(1,2,4)(3,5,8));