Elements of the Group $\langle a,b\mid a^2,b^3,(ab)^8,(ab)^4(ab^{-1})^4 \rangle.$

Question

Consider the group $\langle a,b\mid a^2,b^3,(ab)^8,(ab)^4(ab^{-1})^4 \rangle$ Which is general linear group $GL(2,3)$ by GAP as I tried below

gap> f:=FreeGroup("a","b");
<free group on the generators [ a, b ]>
gap> a:=f.1;;b:=f.2;;
gap> g:=f/[a^2,b^3,(a*b)^8,(a*b)^4*(a*b^-1)^4];;
gap> Size(g);
48
gap> StructureDescription(g);
"GL(2,3)"

Now i want all $48$ elements of the group $\langle a,b\mid a^2,b^3,(ab)^8,(ab)^4(ab^{-1})^4 \rangle$ in the form of $a$ and $b$ only i.e. words in $a$ and $b$ like $\{e,,a,b,b^{2},ab,(ab)^{2},\cdot\cdot\cdot (ab)^{7}\cdot\cdot\cdot\}.$ Please help me to find all $48$ elements of the group. Is there some algorithm in GAP to find out elements. Thank you.

Answer 1

In GAP, you can ask for Elements for any finite group and get (memory permitting) a list of elements. So you could do this for your group g. Caveat: GAP does (for good reasons which I do not want to elaborate here) not by default reduce word expressions. So you will get a nicer result by forcing reduced multiplication (using an ad-hoc length-based rewriting system:

gap> SetReducedMultiplication(g);
gap> Elements(g);
[ <identity ...>, a, b^-1, b, a*b^-1, a*b, b^-1*a, b*a, a*b^-1*a, a*b*a,
  b^-1*a*b^-1, b^-1*a*b, b*a*b^-1, b*a*b, (a*b^-1)^2, a*b^-1*a*b, a*b*a*b^-1,
  (a*b)^2, (b^-1*a)^2, b^-1*a*b*a, b*a*b^-1*a, (b*a)^2, (a*b^-1)^2*a,
  a*b^-1*a*b*a, a*b*a*b^-1*a, (a*b)^2*a, (b^-1*a)^2*b^-1, (b^-1*a)^2*b,
  b^-1*a*b*a*b^-1, b^-1*(a*b)^2, b*(a*b^-1)^2, b*a*b^-1*a*b, (b*a)^2*b^-1,
  (b*a)^2*b, (a*b^-1)^3, (a*b)^3, (b^-1*a)^3, (b^-1*a)^2*b*a,
  b^-1*a*b*a*b^-1*a, b^-1*(a*b)^2*a, b*(a*b^-1)^2*a, b*a*b^-1*a*b*a,
  (b*a)^2*b^-1*a, (b*a)^3, (a*b^-1)^3*a, (a*b)^3*a, (b*a)^3*b, (b*a)^4 ]