Dear Forum, I wish to construct a family of groups parametrised by a variable n, to be specified by the user. Let me describe the steps:
0) Specify a value for n, 1) construct the finite field F=F_q with q=2^(2n+1) elements, 2) construct the automorphism f of F mapping each element x to x^(2^a), where a=n+1, 3) define an operation * on FxF (as a set) via the rule: (x,y)*(x',y')=(x+x',y+y'+xf(x')), where the operations on the right-hand-side are those of the underlying field F, Now P=(FxF,*) is a group, but I don't know if I can assure GAP this is the case. Perhaps I can ask her to test this somehow (GAP is a woman in my world). 4) assuming that GAP now knows P is a group, define the semi-direct product S=P]C, where C is the multiplicative group of F and acts on P (via automorphisms) by the rule:c.(x,y)=(cx,cf(c)y), where f is as in 2) and concatenation on the right-hand-side is just multiplication in F. To give some motivation for the above construction, P thus constructed is the Sylow 2-subgroup of Sz(q), while S is its normaliser in Sz(q). Of course I can access both P and S via P:=SylowSubgroup(Sz(q),2) and S:=Normaliser(Sz(q),P) respectively, but I would prefer to work with P and S constructed as per the above "algorithm". The reason is that I mainly want to count subgroups and conjugacy classes of subgroups of P and S for (at least) n=4, since for smaller values of n, the order of C is a prime (thus C uninteresting), but GAP is unable to handle this in terms of memory. I have serious doubts that constructing P and S as above will be more memory-efficient than the permutation representation already in Sz(IsPermGroup,q), but I can't decide this issue in advance. If it is indeed non-trivial to decide beforehand, I would appreciate any help in how to write the code for the construction as explained above. Many thanks, Stefanos _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
