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Generating p-groups






The p-central series of a group G is the 
descending sequence of subgroups
 G= P_(0)(G) >= ... >= P_(i - 1)(G) >= P_i(G) >= ... >= where P_i(G)=[P_(i - 1)(G), G]P_(i - 1)(G)^p for i >= 1.


If P_c(G)=1 and c is the smallest such integer then G has
p-class c. A group with p-class c is nilpotent and 
has nilpotency class at most c.  


Let G be a finite p-group with Frattini rank d and class c.
A group H is a descendant of G if H has Frattini rank d and
the quotient H/P_c(H) is isomorphic to G. A group is an immediate
descendant of G if it is a descendant of G and has class c + 1.


The p-group generation algorithm allows the construction
of (immediate) descendants of a p-group.
For a description of this algorithm, see
[New77], [O'B90].


SetVerbose ("GeneratepGroups", 1) will provide
information on the progress of the algorithm.



GeneratepGroups (p, d, c : parameters) : RngIntElt, RngIntElt,RngIntElt -> [GrpPC]



Generate all d-generator p-class at most c p-groups.







     Exponent: RngIntElt                 Default: 0




All groups constructed satisfy the supplied exponent.





     OrderBound: RngIntElt               Default: 0




Given OrderBound := n, all groups constructed have order at most p^n.









     StepSizes: [RngIntElt]              Default: null




Construct descendants of order p^((n + s)) of a group of 
order p^n only for s in StepSizes.   





     All: BoolElt                        Default: true




If true, return all groups. Otherwise, return only
the capable groups (those which have descendants).



Descendants(G : parameters) : GrpPC -> [GrpPC]


Descendants(G, c : parameters) : GrpPC, RngIntElt -> [GrpPC]



Construct descendants of G having p-class at most c; if
c is not supplied, it is assumed to be one larger than the
p-class of G.
This function supports the same variable arguments as GeneratepGroups.





Example GrpPGp_Generating_p_groups (H26E1)






> G := DihedralGroup(GrpPC, 16);
> T := Descendants (G, 8);
> #T;
12
> H := T[5]; 
> H;
GrpPC : H of order 128 = 2^7
PC-Relations:
    H.1^2 = H.7, 
    H.2^2 = H.3 * H.4, 
    H.3^2 = H.4 * H.5, 
    H.4^2 = H.5 * H.6, 
    H.5^2 = H.6 * H.7, 
    H.6^2 = H.7, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.1 = H.3 * H.4, 
    H.4^H.1 = H.4 * H.5, 
    H.5^H.1 = H.5 * H.6, 
    H.6^H.1 = H.6 * H.7






Example GrpPGp_GeneratepGroups (H26E2)

What is the soluble length of a 2-generator group of exponent 4?
We construct the 2-generator 2-groups having exponent 4.





> T := GeneratepGroups(2, 2, 10: Exponent := 4);
> "The number of 2-generator exponent 4 groups is ", # T;
The number of 2-generator exponent 4 groups is  26


What are their soluble lengths?





> for i := 1 to #T do
>       "Group ", i, " has soluble length ", DerivedLength (T[i]);
> end for;
Group  1  has soluble length  1
Group  2  has soluble length  2
Group  3  has soluble length  2
Group  4  has soluble length  1
Group  5  has soluble length  2
Group  6  has soluble length  2
Group  7  has soluble length  2
Group  8  has soluble length  2
Group  9  has soluble length  2
Group  10  has soluble length  2
Group  11  has soluble length  2
Group  12  has soluble length  2
Group  13  has soluble length  2
Group  14  has soluble length  2
Group  15  has soluble length  2
Group  16  has soluble length  2
Group  17  has soluble length  2
Group  18  has soluble length  2
Group  19  has soluble length  2
Group  20  has soluble length  2
Group  21  has soluble length  3
Group  22  has soluble length  3
Group  23  has soluble length  3
Group  24  has soluble length  3
Group  25  has soluble length  3
Group  26  has soluble length  3





Example GrpPGp_IsGood (H26E3)

Can we find all 2-generator 3-groups of abundance zero?
Such groups have order at most 3^5.
First, we define a function which checks the number
of conjugacy classes of a group (to determine abundance).





> IsGoodGroup := function(G, k)
>
>    ncl := # Classes(G);
>
>    O := FactoredOrder(G);
>    p := O[1][1];
>    m := O[1][2];
>    n := Floor(m / 2);
>    e := m - n * 2;
>    Desired := n * (p^2 - 1) + p^e + k * (p - 1) * (p^2 - 1);
>
>    return (Desired eq ncl);
>
> end function;


Then, we generate the potential candidates and check
each.





> a := GeneratepGroups (3, 2, 4 : OrderBound := 5); 
> #a;
42 
>
> for i := 1 to #a do
>        G := a[i];
>        if IsGoodGroup(G, 0) then
>           "Group ", i, " of order ", Order(G), " has abundance 0";
>        end if;
> end for;
Group  1  of order  9  has abundance 0
Group  3  of order  27  has abundance 0
Group  4  of order  27  has abundance 0
Group  11  of order  81  has abundance 0
Group  12  of order  81  has abundance 0
Group  13  of order  81  has abundance 0
Group  14  of order  81  has abundance 0
Group  40  of order  243  has abundance 0
Group  41  of order  243  has abundance 0
Group  42  of order  243  has abundance 0





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