# Equivalence Relation on Symmetric Group by Image of n is Congruence Modulo Subgroup

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## Theoerm

Let $S_n$ denote the symmetric group on $n$ letters

Let $\sim$ be the relation on $S_n$ defined as:

- $\forall \pi, \tau \in S_n: \pi \sim \tau \iff \map \pi n = \map \tau n$

Then $\sim$ is an equivalence relation which is congruence modulo a subgroup.

## Proof

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 37 \delta$