Center not is weakly normal-homomorph-containing
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
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.