Center not is image-closed characteristic
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., image-closed characteristic subgroup)
View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions
Clearly, is the center of .
Now consider the quotient map by . Under this quotient map, the group maps to , with mapping to the direct factor in . By our assumption, is isomorphic to , so this quotient group looks like . Clearly, is not characteristic in this.
The particular example can be obtained from the generic example above by setting to be a cyclic group of order 2 and (the symmetric group on three letters). in this case is the subgroup of order three generated by a 3-cycle.