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

http://www.gap-system.org/pkg/xgap/htm/chapters.htm

see in particular, section 4.5 at

http://www.gap-system.org/pkg/xgap/htm/CHAP004.htm#SECT005

	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   

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