[GAP Forum] Wreath Product Decomposition

[Alastair Donaldson]

Dear Forum members

I am trying to write an algorithm which, given a permutation group G 
acting (not necessarily transitively) on {1,2,.....,n} for some n>0, will 
determine whether or not G is a wreath product of subgroups H and K.

According to a previous posting by Burkhard Hoefling, I think this should 
be fairly easy:

"Wreath products of permutation groups can easily be recognized by looking 
at their block structure, see `Blocks' in the GAP reference manual. In
general, a transitive permutation group G embeds in the wreath
product (action of block stabilizer on block) wr (action of G on
orbit of block), and you can easily check equality by comparing orders."

However, the groups which crop up in my application domain do not tend to 
act transitively.  As an example, consider the group

G := Group( [ (1,2), (2,3), (1,4)(2,5)(3,6)(19,20), (26,27), (28,29), 
(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(19,21)(20,22)(25,30)(26,31)(27,32)(28,33)(29,34)(40,41), 
(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(21,23)(22,24)(30,35)(31,36)(32,37)(33,38)(34,39)(41,42) 
]);

I have worked out that this decomposes in several possible ways as H wr K, 
one of which is

H := Group([ (33,34), (31,32), (11,12), (8,9), (10,11,12), (7,8,9), 
(7,10)(8,11)(9,12)(21,22) ]);

and

K := Group([ (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(21,23)(22,24)(30,35)(31,36)(32,37)(33,38)(34,39)(41,42), 
(1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,21,23)(20,22,24)(25,30,35)(26,31,36)(27,32,37)(28,33,38)(29,34,39)(40,41,42) ]

(the group H can then be decomposed further as a direct product, one 
factor of which is itself a wreath product).

I have written a rather complex algorithm which does this decomposition 
correctly for my example.  However, I'm having trouble proving the 
correctness of my algorithm, which is not nearly as simple as Buckhard's approach of looking at blocks 
and comparing orders.  However, since blocks are not immediately 
applicable when the group does not act transitively, I'm not sure how to 
extend his suggested approach.

Any ideas would be greatly appreciated

-Alastair