Every metric $d$ on $\{1,\ldots,n\}$ induces a metric
$D(a,b) := \sum_{i=1}^{n} d(a(i),b(i))$ on the symmetric group $S_n$.
Is this induced metric right or left invariant or both?
Left invariance: $D(c \cdot a, c \cdot b) = D(a,b) $ for all $a,b,c \in S_n$
Right invariance: $D(a \cdot c, b \cdot c) = D(a,b) $ for all $a,b,c \in S_n$
Multiplication: $c := a\cdot b$ is defined as $c(i) = a(b(i))$ for all $i$.