Dihedral group of order 10 in GAP

Question

The dihedral group of order $10$ is given by $D_{10} = \langle a,b| a^5 = b^2 = 1, bab^{-1} = a^{-1}\rangle$. Now I need to find all the elements in GAP. But whenever I type DihedralGroup(10); in GAP the output show the elements $$[\text{<identity> of ...}, f_2, f_2^2, f_2^3, f_2^4, f_1, f_1f_2, f_1f_2^2, f_1f_2^3, f_1f_2^4 ]$$ How do I get the elements in representatives of $a$ and $b$?

Answer 1

GAP is treating the Dihedral Group as a "pc-group" - this is good for GAP, but when you are just learning group theory it might not be so clear for you :D

Here is one option that might help you. We can just build the group with the presentation/letters exactly as you gave in the question.

gap> f:=FreeGroup("a","b");
<free group on the generators [ a, b ]>
gap> g := f/[f.1^5 ,f.2^2, f.2*f.1*f.2^-1*f.1];
<fp group on the generators [ a, b ]>
gap> SetReducedMultiplication(g);
gap> ShowMultiplicationTable(g);

We can double check we have the group we think we do...

gap> IdGroup(g);
[ 10, 1 ]
gap> IdGroup(DihedralGroup(10));
[ 10, 1 ]

Also maybe you want to be able to do some arithmetic in this group, you could do the following:

gap> a:=g.1;;
gap> b:=g.2;;
gap> a*b*a;
b
gap> a^2*b*a*b;
a