Perfect core

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

Definition

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 perfect normal subgroup.
3. The unique largest 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)

In terms of the group property core operator

This property is obtained by applying the group property core operator to the property: perfectness
View other properties obtained by applying the group property core operator

Property theory

Monotonicity

The perfect core subgroup function is monotonic: if $H$ ≤ $G$, 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 invariant subgroup and in fact a homomorph-containing 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 invariant.

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.