Let us say that a non averaging sequence in a group G is a sequence x1,…,xn such that

x2i≠xjxk

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 Z), but I would like to know if there exist more “optimal” results.

**Answer**

**Attribution***Source : Link , Question Author : vizietto , Answer Author : Community*