[GAP Forum] p-group

[Bettina Eick]

Dear Vivek kumar jain,

you can use GAP to investigate your question for any fixed prime p.

For example, the nilpotent quotient algorithm of the NQ package or the
NQL package of GAP allows you to determine the largest class-c quotient 
of a finitely presented groups for any positive integer c or even the
largest nilpotent quotient (if this exists).

Further, there are methods available in GAP to determine the automorphism
group of a finite p-group. Check the AutPGrp package for this purpose.

In your given example, you can implement your considered group G in GAP
as function in p:

G := function(p)
     local F, f, r, a, b, c;
     F := FreeGroup(3);
     f := GeneratorsOfGroup(F); a := f[1]; b := f[2]; c := f[3];
     r := [a^(p^5), b^(p^3), c^(p^2),
           Comm(a,b)/a^(p^3),
           Comm(a,c)/c^p,
           Comm(b,c)/b^(p^2) ];
     return F/r;
end;

Then you load the relevant packages

LoadPackage("nq");
LoadPackage("autpgrp");

And then you can do the following (for example for p=3):

gap> H := G(3);
<fp group on the generators [ f1, f2, f3 ]>
gap> K := NilpotentQuotient(H);
Pcp-group with orders [ 27, 9, 3, 9, 3, 3 ]
gap> Length(LowerCentralSeries(K));
3
gap> A := AutomorphismGroupPGroup(K);;
gap> A.size;
14348907

Hence for p=3 your group has class 2 and you can see the size
of its automorphism group. Generators and further information on
the automorphisms is also stored in A, but is perhaps too long to
be displayed here.

Hope this helps,

Bettina



> "Is it possible using GAP to check that given presentation is a nilpotent group of class 2 or not?"
>
> For example $G=\langle  a,b,c| a^{p^5}, b^{p^3}, c^{p^2}, [a,b]=a^{p^3}, [a,c]=c^p, [b,c]=b^{p^2} \rangle $ where $p$ is a prime.
>
> Also how can we determine its automorphism group using GAP?
>
>
> with regards
>
> Vivek kumar jain
>
>
>
>
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