Let us say that a non averaging sequence in a group $G$ is a sequence $x_1, \dots, x_n$ such that $$x_i^2 \neq x_j x_k$$ for any indices $i, j, k$ such that two at least are distinct. My question is: given $n$, what is the smallest group having a non-averaging sequence of length $n$?
Has this question been already tackled in the literature, maybe under a different name?
There are some partial results (we can build cyclic groups with non-averaging sequence based on the existence of non-averaging sequences in $\mathbb{Z}$), but I would like to know if there exist more "optimal" results.
