# Non averaging sequences in finite groups

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.

• One easy example is $\{0,1\}^n\subset(\mathbb Z_3)^n$. Aug 26, 2015 at 22:01