Perfect core

Symbol-free definition
The perfect core or stable commutator of a group is all of the following equivalent things:


 * 1) The unique largest perfect subgroup.
 * 2) The unique largest defining ingredient::perfect normal subgroup.
 * 3) The unique largest defining ingredient::perfect characteristic subgroup.
 * 4) The subgroup generated by all perfect subgroups
 * 5) The limit of the (possibly transfinite) derived series. For a finite group, or more generally for an Artinian group, this limit is attained at some finite stage.

Definition with symbols
(fillin for all equivalent conditions)

Monotonicity
The perfect core subgroup function is monotonic: if $$H$$ &le; $$G$$, then the perfect core of $$H$$ is contained in the perfect core of $$G$$.

Idempotence
The perfect core of the perfect core is again the perfect core. This is because the perfect core is always a perfect group.

Subgroup properties satisfied by the perfect core
The perfect core of a group is a fully characteristic subgroup, because any image of a perfect group under a homomorphism is again a perfect group.

The perfect core of a group is intermediately characteristic, in fact it is intermediately fully characteristic.

More generally, if the perfect core always satisfies property $$p$$, it also always satisfies the property intermediately $$p$$. This follows from the fact that it is an intermediacy-preserved subgroup function on account of being both monotone and idempotent.