Center not is weakly normal-homomorph-containing

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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., weakly normal-homomorph-containing 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 with center Z and a homomorphism f:Z \to G such that f sends normal subgroups of G contained in Z to normal subgroups of G but f(Z) is not contained in Z.

Facts used

  1. Center not is normality-preserving endomorphism-invariant

Proof

The proof uses Fact (1). In particular, the same generic example works, as does the same particular example of direct product of S3 and Z3.