Center not is normality-preserving endomorphism-invariant
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
It is possible to have a group such that the center is not normality-preserving endomorphism-invariant, i.e., there exists a normality-preserving endomorphism of (i.e., sends normal subgroups to normal subgroups) but is not contained in .
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|
Clearly, (as an embedded direct factor) is the center of .
Now consider the endomorphism of which composes the projection from onto with an isomorphism from to . Note that:
- is normality-preserving: Its image, is a cyclic normal subgroup of a direct factor . Since direct factor implies transitively normal and cyclic normal implies hereditarily normal, all subgroups of the image are normal in . In particular, the image of any normal subgroup in is normal.
- The endomorphism does not send to itself: Rather, it gets mapped to .
Find a group with an abelian normal subgroup .
Modification of generic example for finite p-groups
The generic example outlined above does not work for finite -groups, but the following slight modification does: instead of requiring to be a centerless group, simply require that not be contained in the center of . In this case, the center of becomes where is the center of . We construct the endomorphism in the same way (project to , compose with isomorphism to ) and note that since is not contained in , this endomorphism does not send to within itself.
Particular examples for finite p-groups
- : We can take as cyclic group:Z4 and as dihedral group:D8 to get direct product of D8 and Z4. Alternatively, we can take as cyclic group:Z4 and as quaternion group to get direct product of Q8 and Z4.
- odd: Take as the cyclic group of prime-square order and as semidirect product of cyclic group of prime-square order and cyclic group of prime order.