Let $G=\langle S\rangle$ be a finitely generated group and $A\subseteq G$ be given. For $g\in G$ we denote by $l_S(g)$ length of $g$ with respect $S$.
In my research, for every $g\in G$ I need to find $h\in G$ with minimal length such that $hg\in A$.
Q. Let $s\in S$ be given and for $g\in G$ choose $h\in G$ with minimal length such that $hg\in A$. If $h$ does not start with $s$, what can say about $sg\in G$? i.e. Can I say $h'= hs^{-1}$ has minimal length such that $h'sg\in A$?
Assume $G=A$, then for any $g\in G$ we get $h=e$, so $h$ does not start or end with $s$, but of course $sg$ is already in $A$ so we may choose $h'=e$ and not $s\cdot h =s$ which has longer length (1 and not 0).