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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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