On Mon, Apr 30, 2018 at 03:26:32PM +0000, somaye madani wrote: > Dear Forum, > I am a PhD candidate in the University of Kashan. I am working finite > groups. I need a group G such that G/[G,G^\prime] is a group of order > 20 that can be written as a semidirect product of $Z_4$ by $Z_5$ with > id (20,3) in Gap notation. Any comments will be highly appreciated.
Such group does not exist (for the same reason that G/G' is always abelian). Set H = [G,G^\prime], and S = G/H. Since S is isomorphic to SmallGroup(20,3), you can find elements A,B,C in S such that [A,[B,C]] is not the identity element: gap> S:=SmallGroup(20,3); <pc group of size 20 with 3 generators> gap> IdGroup(S/CommutatorSubgroup(S,DerivedSubgroup(S))); [ 4, 1 ] Now pick cosets representative a,b,c in G such that A = aH, B = bH, C = cH By definition [a,[b,c]] belongs to H, so H = [a,[b,c]]H = [aH,[bH,cH]] = [A,[B,C]]. So [A,[B,C]] is the identity element of S, which contradicts the hypothesis. Cheers, Bill. _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
