Good morning everyone,  I'm looking for a subgroup of $Z(G)$ the center of the group $G$ that is not characteristic subgroup of $G$.  Did someone have that in his pocket ?  I wish you a very good day !
In the book I'm study is written:     A normal subgroup of a group need not be characteristic.   And as an exercise I'm supposed to find an example, it also said that is pretty hard to find one. After trying for two days I wasn't able to find one example.   So, I'm asking for an example of a group $G$ with a normal subgroup $H$ that is not a characteristic of $G$.  I will add some context because maybe it will clarify why the book said it is difficult to find an example: the main problem with the exercise is that until it the book had only covered Group Definition, Subgroups, Lagrange's Theorem and Homomorphisms. So I'm supposed to find a example with such lack of advanced tools.
In the book I'm study is written:     A normal subgroup of a group need not be characteristic.   And as an exercise I'm supposed to find an example, it also said that is pretty hard to find one. After trying for two days I wasn't able to find one example.   So, I'm asking for an example of a group $G$ with a normal subgroup $H$ that is not a characteristic of $G$.  I will add some context because maybe it will clarify why the book said it is difficult to find an example: the main problem with the exercise is that until it the book had only covered Group Definition, Subgroups, Lagrange's Theorem and Homomorphisms. So I'm supposed to find a example with such lack of advanced tools.