[GAP Forum] PQuotients - meaning???

[Werner Nickel]

Dear Forum,
dear Michael,

> I am trying to find all groups of order 2^8 which are quotients of a 
> certain infinite FP group.

this might actually be a fairly difficult problem if the infinite
group has lots of quotients of that order.

> I know I could use GQuotients combined with SmallGroup, but checking 
> *every* small group
> of order 256 takes a long time. And 2^8 is actually a testbed for the 
> real problems - 2^9 and 2^10.

For the fp-group below you would only need to look at groups with
Frattini factor of order 2^3.  There are 6190 such groups among the
56092 groups of order 2^8.
 
> I read up on Quotient Methods, and got a feeling the PQuotient may be 
> the method I need.
> 
> I tried the following -
> 
> gap> F := FreeGroup("a","b","c");
> <free group on the generators [ a, b, c ]>
> gap> a := F.1;; b := F.2;; c := F.3;;
> gap> rels := [a^2, b^2, c^2, (a*c)^2];;
> gap> W := F/rels;
> <fp group on the generators [ a, b, c ]>
> gap> PQuotient(W,2,8);
> <2-quotient system of 2-class 8 with 55 generators>
> 
> Can someone help me understand what all this means? 
> Why is my quotient 
> of order 2^55, and what does that have to do with the parameter '8' I 
> passed to PQuotient? 

You have computed the largest 2-quotient of W with 2-class 8.
The p-central series of a group G is defined by [G_i,G]G_i^p  with
G_1=G.  The p-class c of G is defined by G_c > 1 and G_{c+1}=1.

You get the largest quotient of 2-central class 8 of W by factoring
out W_9.  This is what the p-quotient computation did.

> Is this order 2^55 group a covering group for *all* 
> the Quotients of the form I want  (this would be nice), 

Yes, it is.  But:

As a group of order 2^8 has 2-class at most 7, you could do

gap> PQuotient(W,2,7);
<2-quotient system of 2-class 7 with 38 generators>

and get a group of order 2^38 (Each generator gives a factor of 2).
In your case, the Frattini quotient has order 2^3, so the maximal
2-class is at most 6:

gap> PQuotient(W,2,6);
<2-quotient system of 2-class 6 with 26 generators>

All quotients of order 2^8 of W are a factor group of this group.


>                 Finally, can PQuotient be wrangled to give what I want - 
> a list of all the quotients of order 256 of my FP group W?

There is an algorithm called p-group generation which can be used to
generate all p-groups with certain properties for given p.  In your
case you want to construct all groups of order 256 that satisfy a
given set of relations. I don't see right now how the existing
machinery can be used to do this.  But I am fairly confident that we
can setup some code to do this.

With kind regards,
Werner Nickel.

-- 
   Dr (AUS) Werner Nickel         Mathematics with Computer Science
   Room:  S2 15/212                    Fachbereich Mathematik, AG 2
   Tel:   +49 6151 163487                              TU Darmstadt 
   Fax:   +49 6151 166535                       Schlossgartenstr. 7
   Email: nickel at mathematik.tu-darmstadt.de       D-64289 Darmstadt
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