Dear all Suppose I have a group algebra FG, where F is the field of order p and G is dihedral group of order 2p. If V(FG) denotes the set of all invertible elements of FG with augmentation 1, then it is known that |V(FG)| = |F|^(2(p-1)) (|F|-1).
I want to find a normal subgroup N of V(FG) such that V(FG) is the split extension of G by N. In case of p=3, I followed the following method: I found all normal subgroups of V(F3D6) of order |V|/|D_6| and then for each N, I found its automorphism group (Aut(N)). Finally a non-trivial homomorphisms in "AllHomomorphismClasses(G, Aut(N))" helped me getting V(F3D6) as split extension of D6. The following is the code I used: p:=3;; g:=DihedralGroup(6);; f:=GF(3);; fg:=GroupRing(f,g);; e:=Identity(fg);; m:=MinimalGeneratingSet(g);; l:=List(m,x-> x^Embedding(g,fg));; g1:=List(g,x-> x^Embedding(g,fg));; u:=Units(fg);; s:=Filtered(u, x-> Augmentation(x) = (Z(p)^(0)) );; v:=AsGroup(s);; h:=Subgroups(v);; h1:=Filtered(h, x-> Size(x) = 27);; h2:=[];; for i in h1 do if IsNormal(u, i) then #Print(i, "\n"); Append(h2, [i]); fi; od; #Print(h2, "\n"); for i in h2 do #Print(l[2] in i, "\n"); od; t:=h2[5];; #gen:=GeneratorsOfGroup(t);; #Print(gen, "\n"); au:=AutomorphismGroup(t);; hom:=AllHomomorphismClasses(g, au);; #Print(Size(au) , "\n"); for i in hom do sdp:=SemidirectProduct(g,i,t);; Print(IsomorphismGroups(v, sdp), "\n"); od; But when I tried the same procedure for p=7 (in order to check whether my conjecture is right or not in general), GAP is not responding at all. Is there some alternate way to check my conjecture? Many thanks in advance. -- *Regards* *Surinder Kaur* *Research scholar * *Department of Mathematics * *IIT Ropar* _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
