[GAP Forum] (no subject)

[Bill Allombert]

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.