Sopsku <rrburns@...> 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 _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum