[GAP Forum] Identifying a large Fp group

[Paul Timmons]

William (cc forum),

I used Monoid Automata Factory (MAF) to simplify your presentation for XX -
see below.

This eliminates the generators f2, f4 and y - and gives a slightly less
unwieldly presentation with
40 relators (assuming no errata in the converting back and forth between
MAF format and
GAP - I did this quite hurriedly).

However this is not amenable to low index subgroup or PQuotient techniques
- even typing
*IsFinite(XX_Simp)* command runs out of memory. So of limited help possibly
but must surely be
preferable to working with XX.

F:=FreeGroup("f1","f3","x");f1:=F.1;f3:=F.2;x:=F.3;

rels:=

[ f3^3, (f3*f1)^2, (x*f1)^3, x^7, f1^8, (f1^3*f3^-1)^2, (f1^3*x)^2,
(x^-1*f1)^4, (x^-1*f3*x^-1*f1)^2,
  f1^3*(f3*f1^-1)^3*f3, f1^2*f3*x^-3*f1*x^2*f3^-1, (x*f3^-1*x^-2*f1)^2,
f1*f3*f1^-1*f3*x^-1*f1*x*f1*x^-3,
  f3^-1*x*f1^-1*x*f3^2*x*f1^-1*x, f1^4*x^-2*(f1^-1*x)^2*x^2, (f1^3*x^-1)^3,
(f1^2*x*f1^-1*x^2)^2,
  f3^-1*f1*x*f1^-1*x^-2*f1*x*f1*x^-1*f3^-1*f1, (f3^-1*x*f3^-1*f1)^3,
(f3^-1*x^-1*f3^-1*f1)^3,
  f1^2*f3*f1^-1*f3*x*f1*x^-1*f1^-2*x^-3,
f1^2*(f1*x^-1)^2*f3^-1*x*(x*f1^-1)^2*x^-1*f3,
  f1^2*f3*x^-1*f1^-1*x*f1^2*x^2*f1*x^-1*f1*f3^-1,
f1^2*f3*x^-2*f1^-2*x^-1*f3^-1*x*f1^-2*x^2,
  f3*f1^-1*x^-1*f1*x^-1*f3^-1*f1*(x^-1*f1*x^-1)^2*f1,
f1^3*x^-1*f3*x^-2*f3*f1^-1*f3*x*f3^-1*x^-3,
  f1^2*x^2*f1^-1*f3^-1*x^-2*f3*f1^2*f3^-1*x^-1*f3^-1*x,
f1*f3^-1*f1*x*f3*f1^-2*f3^-1*x*(f3^-1*f1)^2*x*f3^-1,
  f1^2*x*f1*x^-1*f1*f3^-1*x^-1*f1^-1*x*f1*x^-1*f1*f3^-1*f1*x,
f1^2*x^2*f1^2*x*f1*x^-1*f1^-1*x^2*f1^-1*x^-1*f1*x,
  f1*(f1*x^-1*f1^-1*x)^3*f1*x^-2,
f1^3*x*(x*f3*f1^-1)^2*f3*f1*x^2*f3^-1*f1^-1*x^-1,
  f1^2*x^3*f1^-1*f3*x^-2*f3^-1*f1*f3^-1*x*f1^-1*f3*f1*x^-1,
  f1*f3*x^-1*f3*f1^-2*f3^-1*x^-1*f3*f1^-1*f3*x^2*f1^-1*x*f3^-1*f1,
  f1^2*x*f1*x^-2*f1^2*f3^-1*x^-1*f1^-1*x^2*f1^-1*f3*f1^-1*x^2,
f1*x^-1*f1^2*x^-3*f1^-1*x^-1*f1*x^-1*f3^-1*f1*f3^-1*x,
  x^-1*f1*x^3*f1^-2*x*f3*x^-1*f1^-1*x*f1*x^-1*f3^-1,
(f1*x^-1*f1)^3*x^2*f1*x^-1*f1*x,
  x^-1*f1*x^-1*f3^-1*x^-1*(f3*f1^-1)^2*x*f3^-1*x^-1*f1*x^-1*f3^-1*x*f3*f1,

x^-1*f1^-2*x^-2*f1^-1*f3^-1*x^2*f1^-2*x*f1*f3*x^2*f1^-1*x^-1*f1*f3*x^-1*f1^-1*x*f3^-1*f1^-2*x*(f3*x^-1)^2*f1^-2*x^-1*f3*x^2*f1^-1*x^-1*f3^-1*f1
];

XX_Simp:=F/rels;
IsPerfect(XX_Simp); # very basic sense check should be "true"


On Wed, 4 Sep 2019 at 18:32, William Giuliano <williamgiuliano00 at gmail.com>
wrote:

> Dear Forum,
>                      I have constructed two finitely presented groups XX
> and YY (attached file) which I am trying to “identify”, or at least to
> understand if they are trivial or not. In the paper “Application of
> Computational Tools for Finitely Presented Groups” by GEORGE HAVAS AND
> EDMUND F. ROBERTSON the authors say that in the case of a perfect group
> (both XX and YY are perfect)
>
> “There are no immediately obvious finite quotients to consider. Suffice it
> to say that finite quotients may exist which are (direct products of
> nonabelian) simple groups. Various techniques exist for finding such
> quotients. There are low index subgroup programs which can sometimes find
> such quotients and have been used to do so. Alternatively, random
> coincidence procedures are also often effective.”
>
> As for YY, I know that it is non trivial, as I managed to find (in MAGMA)
> surjective homomorphisms from YY to G2(3), and therefore some subgroups U
> of finite index for which I computed U/U’ and U’/U”. But apart from that, I
> can’t say more.
>
> As for XX - which I’m mainly interested in - I tried with LowIndexProcess
> (in MAGMA) as well as with Index(XX,Subgroup(XX,[XX.1,XX.2,XX.3,XX.4]));
> but
> it seems it takes too long to give an answer.
>
> Do you know any methods I could use in practice to see at least if XX is
> trivial or not?
>
> Thank you very much,
> William
> _______________________________________________
> Forum mailing list
> Forum at gap-system.org
> https://mail.gap-system.org/mailman/listinfo/forum
>