Non-roots of unity auxillary constants in a group?

Question

Let $A$ be a set, together with a set $F$ of n-ary operations on A, which may include constants of $A$ as 0-ary operations. A set $G$ of operations on $A$ is said to be auxillary with respect to the algebra $(A,F)$ if $G$ is disjoint from $F$ and the equational identities of $(A,F)$ generate the equational identities of $(A, F \cup G)$. For example, in $(\mathbb R, +)$, every {$r$} where $r$ is a nonzero real, is an auxillary set. My question is, take a group $G$ in the signature {$*$}. Suppose $g$ is not an n-th root of unity. Is {$g$} an auxillary set with respect to $(G, *)$. I apologize if the question is too long, but I would be VERY surprised to see a counterexample.