co.combinatorics - A number theoretic identity

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

$$ S_lambda = sum_{a=1}^n |a-(alambda)'|, $$

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 $lambda geq 2$?