Center not is normality-preserving endomorphism-invariant

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) does not always satisfy a particular subgroup property (i.e., normality-preserving endomorphism-invariant subgroup)
View subgroup property satisfactions for subgroup-defining functions | View subgroup property dissatisfactions for subgroup-defining functions

Statement

It is possible to have a group G such that the center H = Z(G) is not normality-preserving endomorphism-invariant, i.e., there exists a normality-preserving endomorphism \alpha of G (i.e., \alpha sends normal subgroups to normal subgroups) but \alpha(H) is not contained in H.

Related facts

Similar facts

Opposite facts

Below are some properties weaker than being normality-preserving endomorphism-invariant, that the center in fact satisfies:

Property Meaning Proof that center satisfies the property
strictly characteristic subgroup invariant under all surjective endomorphisms center is strictly characteristic
finite direct power-closed characteristic subgroup finite direct power of subgroup closed in corresponding power of whole group center is finite direct power-closed characteristic
direct projection-invariant subgroup invariant under projections to direct factors center is direct projection-invariant
characteristic subgroup invariant under all automorphisms center is characteristic

Proof

Generic example

Let A be a nontrivial cyclic group and C a centerless group containing a normal subgroup isomorphic to A, say B. Consider the direct product G := A \times C.

Clearly, H := A \times \{ e \} (as an embedded direct factor) is the center of A \times C.

Now consider the endomorphism \alpha of G = A \times C which composes the projection from G onto A with an isomorphism from A to \{ e \} \times B. Note that:

Find a group A with an abelian normal subgroup B.

Particular example

The above generic example can generate a particular example by setting A as cyclic group:Z3 and C as symmetric group:S3. The group G is thus direct product of S3 and Z3.

Modification of generic example for finite p-groups

The generic example outlined above does not work for finite p-groups, but the following slight modification does: instead of requiring C to be a centerless group, simply require that B not be contained in the center of C. In this case, the center of G = A \times C becomes H = A \times Z(C) where Z(C) is the center of C. We construct the endomorphism in the same way (project to A, compose with isomorphism to \{ e \} \times B) and note that since B is not contained in Z(C), this endomorphism does not send H to within itself.

Particular examples for finite p-groups