[GAP Forum] D8 as a product of Q=D8/Center and its Center

[Sopsku]

Sopsku <rrburns at ...> writes:
> 
> Dear Forum
> 
> I am tryng to use GAP to "reconstruct" the group D8 as some sort of product
> of its quotient group Q=D8/Center(D8) and its Center(D8).
> 
> gap> D8:=DihedralGroup(8); # or however best defined for this
> <pc group of size 8 with 3 generators>
> gap> ZD8:=Center(D8);
> Group([ f3 ])
> gap> Q:=D8/ZD8;
> Group([ f1, f2, <identity> of ... ])
> gap> StructureDescription(Q);
> "C2 x C2"
> 
> I know it can't be direct product. I reorganized the Cayley Graph for D8 by
> grouping the cosets as nodes and and looking at the lines connecting the
> nodes. They cross and so a direct product will not do it. Verify this:
> 
> gap> d:=DirectProduct(Q,ZD8);
> <pc group of size 8 with 3 generators>
> gap> StructureDescription(d);
> "C2 x C2 x C2"
> 
> Not a direct product. See if I can make a semidirect product
> 
> gap> A:=AutomorphismGroup(ZD8);
> <group of size 1 with 2 generators>
> gap> List(A,Order);
> [ 1 ]
> 
> I don't see how to match up any orders and proceed from here.
> 
> I would appreciate any help as how to use GAP functionality to construct 
> this product Thank you for any help.
>       Ron
> 

Dear Forum,

In my original pointing I made a mistake in stating the I could not
"reconstruct" the group D8 as some sort of product of its quotient group
Q=D8/Center(D8) and its Center(D8). In fact (since Q iso C2XC2 and
Center(D8) iso C2) this is simply done as follows:

gap> C2 := Group( (1,2) );;
gap> C2xC2 := Group( (1,3)(2,4), (1,2)(3,4) );;
gap> 
gap> A:=AutomorphismGroup(C2xC2);;
gap> L:=Elements(A);;
gap> List(L,Order);
[ 1, 2, 2, 3, 3, 2 ]
gap> phi:=GroupHomomorphismByImages(C2,A,[C2.1],[L[2]]);
[ (1,2) ] -> [ [ (1,3)(2,4), (1,2)(3,4) ] -> [ (1,4)(2,3), (1,2)(3,4) ] ]
gap> s:=SemidirectProduct(C2,phi,C2xC2);
Group([ (3,4)(5,6), (3,5)(4,6), (1,2)(5,6) ])
gap> StructureDescription(s);
"D8"

Which is what I wanted to see. As pointed out to me by Attila Egri-Nagy I
can also construct D8 as a wreath product from my C2's

gap> w:=WreathProduct(C2, C2);
Group([ (1,2), (3,4), (1,3)(2,4) ])
gap> StructureDescription(w);
"D8"

but this does not show D8/Center(D8) X Center(D8) as explicitly as the
semidirect product shows it.

I apologize for the mistake in my original posting and thank you all for
your consideration.
  Ron