An element $s$ of a group $G$ is a logical generator of $G$ iff every element of $G$ can be defined in the first order language of groups with $s$ as a parameter. In this case we may call $G$ a logically cyclic group. This is equivalent to say that for any elementary extension $G^{\ast}$ of $G$ and any automorphism $\alpha:G^{\ast}\to G^{\ast}$, the condition $\alpha(s)=s$ implies $\alpha_{|_G}=\mathrm{id}$.
As a result, the set $\{ s\}$ is an $\mathrm{Aut}$-basis of $G$ in terms of [1] (i.e., every automorphism of $G$ is uniquely determined by its value on $s$).
I started to study logically cyclic groups in [2]. Recently, I realized that in fact I don't have any example of a group which is not logically cyclic but it has a singleton $\mathrm{Aut}$-basis. So, here is my question:
Is it true that if $\{ s\}$ is an $\mathrm{Aut}$-basis of $G$, then $s$ is a logical generator of $G$?
[1] G. Cutolo, C. Nicotera: Subgroups defining automorphisms in locally nilpotent groups, Forum Mathematicum, 15 (2003).
[2] M. Shahryari: On logically cyclic groups, J. Group Theory, 18 (2015).
Best Answer
$\{1\}$ is an Aut-basis of the group of profinite integers $\hat{\mathbb Z}$, but since this group has cardinality $2^\omega$, it has no logical generator. However, every element of $\hat{\mathbb Z}$ is type-definable from $1$ (i.e., any pair of distict elements are distinguishable by a first-order formula with parameter $1$); using Ali Enayat’s terminology, the structure $\langle\hat{\mathbb Z},+,1\rangle$ is Leibnizian. (Conversely, if $G$ is the group reduct of a $\mathbb Z$-group such that $\{1\}$ is an Aut-basis of $G$, then all elements of $G$ are type-definable from $1$, and $G$ embeds in $\hat{\mathbb Z}$; see my paper Rigid models of Presburger arithmetic.)
Similarly, if $p$ is any prime, the group of $p$-adic integers $\mathbb Z_p$ has no logical generator (being uncountable), but all its elements are type-definable from $1$, thus $\{1\}$ is an Aut-basis.
Note that “every element of $G$ is type-definable from $s$” gives you a general property to consider that is intermediate between “$\{s\}$ is an Aut-basis of $G$”, and “$s$ is a logical generator of $G$”.