# Perfect groups, central extensions and abelian groups

abelian-groupsfinite-groupsgroup-extensionsgroup-theory

Let $$G$$ be a perfect group. We often consider extensions
$$1 \to A \to E \to G \to 1,$$

where $$A$$ is abelian, $$G$$ is perfect and the image of $$A$$ is central. Is it possible for such an extension not be central? In other words if we have
$$1 \to A \to E \to G \to 1,$$
where $$A$$ is finite abelian, $$G,E$$ finite perfect then must the image of $$A$$ be central?

Let $$W := \mathbb{Z}_2 \wr A_5$$ and let $$G := \operatorname{Inn}(W)$$. This is a perfect group of order $$960$$, has trivial centre, and is of the form $$(\mathbb{Z}_2)^4 \rtimes A_5$$.

The following GAP code can identiy more examples:

n := 5000;
for i in [ 1..n ] do
for j in [ 1..NrPerfectGroups( i ) ] do
G := PerfectGroup( i, j );
Ns := Filtered( NormalSubgroups( G ), N -> IsAbelian( N ) and not IsSubgroup( Centre( G ), N ) );
if not IsEmpty( Ns ) then
Print( "PerfectGroup( ", i, ", ", j, " ) is an example.\n" );
fi;
od;
od;