Does there exist a group with trivial center and non-trivial norm?

Question

Let’s call the norm of a group $G$ its subset $N(G) = \{g \in G\mid \forall H < G,\ gHg^{-1} = H\}$ (all subgroups are closed under conjugation by elements of norm). It is true, that $N(G)$ is a characteristic subgroup of $G$, and that it contains $Z(G)$. However, $N(G)$ is sometimes larger, than $Z(G)$. For example $N(Q_8) = Q_8$, but $Z(Q_8) \cong C_2$.

My question is:

Does there exist such a group $G$, such that $Z(G) \cong E$, but $N(G)$ is non-trivial?

I failed to find any, but maybe I just searched in a wrong place…

Answer 1

Such groups indeed do not exist.

A theorem proved by C.D.H. Cooper in "Power automorphisms of a group" states:

Any automorphism that leaves every subgroup of a group $G$ invariant induces the trivial automorphism in $\frac{G}{Z(G)}$.

This fact implies, that centerless groups do not have non-trivial power automorphisms. Particularly, that means, that if $Z(G)$ is trivial, then so is $N(G)$.