[GAP Forum] Double Semidirect Product Group

[Stephen Linton]

Dear GAP Forum:


On 19 Feb 2011, at 14:16, Sandeep Murthy wrote:

> Hi,
> 
> I have written a GAP function that constructs double semidirect product groups of type
> 
> ((C_l)^2 \rtimes_s C_m) \rtimes_t C_n
> 
> where C_l, C_m, C_n are cyclic groups of orders l, m, n respectively, and C_m
> acts on (C_l)^2 via a nontrivial homomorphism s: C_m --> Aut((C_l)^2), and
> C_n acts on the semidirect product (C_l)^2 \rtimes_s C_m via a nontrivial
> homomorphism t: C_n --> Aut((C_l)^2 \rtimes_s C_m)).  So for example for 
> l = 5, m = 3, n = 2, it should give a semidirect product group of order 150, one of
> [150, 5], [150,6], [150,9] depending on the choices of homomorphisms.
> 
> However, when I run it on GAP I get an error:
> 
> Error, usage: Image(<map>), Image(<map>,<elm>), Image(<map>,<coll>) called from
> Image( aut, pcgsG[i] ) called from
> SplitExtension( G, aut, N ) called from
> <function>( <arguments> ) called from read-eval-loop
> Entering break read-eval-print loop ...
> you can 'quit;' to quit to outer loop, or
> you can 'return;' to continue
> brk> 
> 


The problem is simply that you have the arguments to SemidirectProduct in the wrong order. The first argument should be the acting group, and the third one the group acted on. The relevant part of the documentation is this:

	

> SemidirectProduct( <G>, <alpha>, <N> )                                 O

constructs  the  semidirect  product  of  <N> with <G> acting via <alpha>.
<alpha>  must  be a homomorphism from <G> into a group of automorphisms of
<N>.

If  <N>  is  a group, <alpha> must be a homomorphism from <G> into a group
of automorphisms of <N>.


	Steve