I don't know whether this answers the more general question which
you recently posed on the GAP forum but it produces two permutations u,
v on an appropriate interval [1..m] which generate the group which you
describe.
gap> a := x -> [x[1],x[1]*x[2]];;
gap> b := x -> [x[2]*x[1],x[2]];;
gap> g := SymmetricGroup(4);;
gap> omega := Cartesian(g,g) ;;
gap> u := PermListList(omega, List(omega,a));;
gap> v := PermListList(omega, List(omega, b));;
gap> h := Group(u,v);;
gap> Size(h);
2239488
It is rather intriguing how the final group h varies with the input g.
- John Dixon
On 2009-11-30 5:22 PM, R. Keith Dennis wrote:
Hi, I'm interested in studying certain groups which arise as
permutations of subsets of a given group. As a simple example, let G
be any finite group, and S = G x G, the set of pairs of elements of G.
Define a to be the permutation of S generated by (x,y) --> (x,xy) and
b to be the one generated by (x,y) --> (yx,y). Is there a simple way
to construct the subgroup of the permutation group on S generated by a
and b? In this case both a and b have order exp(G) & for example I'd
like to be able to compute a presentation for the group of
permutations they generate. More generally I'd like to study other
operations on certain subsets (or sequences) of elements derived from
a fixed group G.
As permutation groups seem to be given as permutations of sets of
integers, it almost seems that I should (in essence) have to describe
a one-to-one correspondence of S with a set of integers [1..m] and
describe a and b by explicitly computing via this correspondence.
Is that sort of thing really necessary? It usually seems that there
are built-in operations in GAP to avoid such.
Perhaps I'm missing something obvious. Suggestions most welcome!
Keith
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