# Identifying $\langle r,s\mid srs^{-2}r, r^{-1}srsr^{-1}\rangle$.

What group is $$G:=\langle r,s\mid srs^{-2}r, r^{-1}srsr^{-1}\rangle?$$

Thoughts . . .

Using IdGroup in GAP on G with

F:=FreeGroup(2);

rels:=[(F.2)*(F.1)*(F.2)^(-2)*(F.1), (F.1)^(-1)*(F.2)*(F.1)*(F.2)*(F.1)^(-1)];

G:=F/rels;


one gets [24, 3], meaning it's the third group in the library, of order $24$. (Where do I find that?)

How would one identify the group independently of GAP?

• $[24,3]$ is $SL_2(\mathbb{F}_3)$. You can look up it here: groupprops.subwiki.org/wiki/Groups_of_order_24, although the site is not officially related to GAP. – pisco Nov 18 '17 at 14:42
• You can find the group with ${\mathtt{SmallGroup}}(24,3)$. ${\mathtt{StructureDescription}}(G)$ will tell you that it is isomorphic to ${\rm SL}(2,3)$. – Derek Holt Nov 18 '17 at 14:57
• It seems to me like you are starting to learn GAP - let me point you to codima.ac.uk/school2016 and alex-konovalov.github.io/gap-lesson which may be useful, and I suggest to watch for our training events in the UK next year. – Alexander Konovalov Nov 19 '17 at 1:24

This is a GAP session which tells that this is $$SL(2,3)$$:

gap> F:=FreeGroup("r","s");
<free group on the generators [ r, s ]>
gap> G:=F/ParseRelators(GeneratorsOfGroup(F),"srs^-2r=r^-1srsr-1=1");
<fp group on the generators [ r, s ]>
gap> Size(G);
24
gap> IdGroup(G);
[ 24, 3 ]
gap> StructureDescription(G);
"SL(2,3)"


It first asks what is the order of the group - StructureDescription is intended primarily for small groups, as its documentation says, so a good idea is to check that the group is "small" first. Also, it may point out to an error in the given presentation, if GAP states that the order is different from what you expect.