Dear GAP Forum,
Juergen Ecker asked:
I have tried to work with the group< p,q,r; p^4, p^2=q^3=(pq)^5, r^2=(rp)^4=p^2 >
in GAP. According to Zassenhaus' "Ueber endliche Fastkoerper" (On finite
nearfields), this is a group having <p,q>=SL(2,5) as a normal subgroup
of index 2.
Nevertheless, I could not make GAP compute the correct size, neither
with the built in coset enumerator, nor using the ace package. The size
of the subgroup <p,q> is computed correctly.
Have I misunderstood Zassenhaus' paper or is there another way of
computing the size of this group in GAP?
I believe that Juergen must have mistranscribed the presentation or
misunderstood the paper in some way.
The subgroup generated by
p, r, q^-1*p*r^-1*q^-1, q^-1*r*p^-1*q^-1, q*p*q*p^-1*q^-1, for instance,
has index 5 and abelian invariants 0,2,2,3 and so is infinite.
This sort of investigation is very easy to carry out using the
GraphicSubgroupLattice command of the xgap package, which is described in the
xgap manual, which is available on line at
see in particular, section 4.5 at
Steve Linton -- Steve Linton School of Computer Science & Centre for Interdisciplinary Research in Computational Algebra University of St Andrews Tel +44 (1334) 463269 http://www-theory.dcs.st-and.ac.uk/~sal Fax +44 (1334) 463278