Let $S_3$ be the symmetric group on the three objects $x_1, x_2, x_3$.
We are given a (countably infinite) sequence of permutations in $S_3$:
$\sigma_1 \ , \ \sigma_2 \ , \ \sigma_3 \ , \ \ldots \ , \ \sigma_n \ , \ \ldots \ \, .$
Is it true that there exists a positive integer $N$ such that the sequence of products
$\sigma_{N} \ , \ \sigma_{N+1}\sigma_{N} \ , \ \sigma_{N+2}\sigma_{N+1}\sigma_{N} \ , \ldots \, ,$
has a(n infinite) subsequence whose terms fix $x_1$? Thanks!!!