Let $n$ be a positive integer such that $2n+1$ is prime. The elements of the factor group $G = \mathbb{F}^\times_{2n+1}/\{\pm 1\}$ can be represented by the integers $1,2,\ldots,n$. For every $x \in \mathbb{F}^\times_{2n+1}$, let $x' \in \{1,2,\ldots,n\}$ denote the representative of its image in $G$. For every $\lambda \in\{2,\ldots,n-1\}$, let

$$ S_\lambda = \sum_{a=1}^n |a-(a\lambda)'|, $$

where $|\cdot|$ denotes the usual absolute value. For example, when $n=6$, and $\lambda=3$, $$ S_3 = |1-3| + |2-6| + |3-4| + |4-1| + |5-2| + |6-5| = 14. $$

Is it true that $S_\lambda = n(n+1)/3$, for every $\lambda \geq 2$?