Re: Standard wreath product & representations

On 20 Oct, 16:55, newsgr.m...@xxxxxxxxx wrote:
Hello,
I have to put the _standard_ wreath product of C_3 by C_2 into a
Z_3C_2-module.

I do not really know what you mean. What exactly do you mean by
putting a group into a module?

Assuming you are thinking of Z_3 as a finite field and C_2 as a cyclic
group of order 2, you can define a Z_3C_2-module of dimension 2, by
defining the action of the generator of C_2 on the vector space to be
the matrix [[0,1],[1,0]].

If you do that, then the semidirect product of the module by the group
will be isomorphic (as a group) to the standard wreath product of C_3
by C_2.

Derek Holt.


I know that this can be done via coniugation by the
elements of C_2, but exactly how? (Which is the representation
needed?)
[please answer here if the question is clear, continue below if
not :D]

---
Let {1,a,a^2} and {1,b} the underlying sets of (respectively) C_3 and
C_2. The basis group of W = C_3 wr C_2 is C_3(1) x C_3(b), so W is

(C_3(1) x C_3(b)) ><| C_2*.

For example, a(1) is the permutation of Sym(C_3 x C_2)

(x,1) |-> (xa,1)
(x,b) |-> (x,b) ;

instead a permutation of C_2* appears like

(x,y) |-> (x,yp)

where p=1* or b*, x in C_3.
The underlying set of B is

{(1(1),1(b)), (1(1),a(b)), (1(1),a^2(b)), ... , (a^2(1),a(b)),
(a^2(1),a^2(b))}.

Obviously, by the definition of wreath product, there is an
automorphism of C_2* in AutB given by

p* |-> ( (c,d) |-> (c,d)^p* ). (*)

In view of a possible Z_3C_2-module representation, we have to look
for a representation r: C_2 -> GL(B). B is a vector space over Z_3, so
that has some sense. Can we assume that r is just a little
modification about (*), taking the domain as C_2 and not C_2* ?

Thanks in advance.

.