I was reading about the Cauchy–Davenport inequality and other results in combinatorics and the following property came to my mind.
Let $G$ be a finite group. I call here $S=\{s_1,\ldots,s_n\}\subseteq G$ a $\pm$-subset for $G$, if
- the identity of $G$ is not contained in S and
- any element $g$ of $G$ can be written as $g=s_{\sigma(1)}^{\varepsilon_1}\cdots s_{\sigma(n)}^{\varepsilon_n}$, where $\sigma$ is a permutation of $\{1,\ldots,n\}$ and each $\varepsilon_i\in\{-1,1\}$.
For example, $\{\bar{1},\bar{2},\bar{3}\}$ is easily seen to be a $\pm$-set for the cyclic group $\mathbb{Z}_5$. On the other hand, it is easy to see that for instance the cyclic group of order $2$ or $4$ do not have any $\pm$-subset. Finally, we may call minimal a $\pm$-subset if it does not contain properly any other $\pm$-subset for the same group.
I have looked for something like this in the literature, but I did not find anything.
Is something known about (minimal) $\pm$-subsets for cyclic groups of odd order?