# 0n group that have an non-trivial element fix with each automorphism

Let $G$ be a group and $Aut(G)$ is an automorphisms groups of $G$. We know that if $Aut(G)$ is nilpotent and $G$ is not cyclic of odd order, then $G$ has an non-trivial element such that fix by all automorphisms. Also it is clear that if $G$ has a characteristic subgroup of order 2, then $G$ has an non-trivial element such that fix by all automorphisms. Now do there exists a group such that $Aut(G)$ is not nilpotent and $G$ has no characteristic subgroup of order 2, but $G$ has an non-trivial element such that fix by all automorphisms?

Thank you

• Casn you give an example of a non-cyclic group $G$ in which ${\rm Aut}(G)$ is nilpotent, and there is no characteristic subgroup of order $2$? If so, then I might be able to construct one in which ${\rm Aut}(G)$ is not nilpotent. – Derek Holt Jun 25 '14 at 13:39
• The group must be of odd order. Let us that i think on it. – elham Jun 25 '14 at 13:43

$G = \mathtt{SmallGroup}(729,31)$ is an example of a group with nilpotent automorphism group $A$. (In fact $|A|=3^9$.) $Z(G)$ is cyclic of order $9$, and an element $z$ of order $3$ in $Z(G)$ is fixed by all $\alpha \in A$.
Now let $H$ be a non-solvable group with centre of order $3$, such as the $3$-fold cover $3.A_6$ of $A_6$, and let $X$ be the central product of $H$ and $G$, amalgamating central elements of order $3$. Then ${\rm Aut}(X)$ is not nilpotent, but the central element of order $3$ in $X$ is fixed by all automorphisms of $X$.