# Definition:Disjoint Permutations

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## Definition

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

Let $\pi, \rho \in S_n$ both be permutations on $S_n$.

Then $\pi$ and $\rho$ are **disjoint** if and only if:

- $(1): \quad i \notin \Fix \pi \implies i \in \Fix \rho$
- $(2): \quad i \notin \Fix \rho \implies i \in \Fix \pi$

That is, each element moved by $\pi$ is fixed by $\rho$ and (equivalently) each element moved by $\rho$ is fixed by $\pi$.

That is, if and only if their supports are disjoint sets.

We may say that:

- $\pi$ is
**disjoint from $\rho$** - $\rho$ is
**disjoint from $\pi$** - $\pi$ and $\rho$ are
**(mutually) disjoint**.

Note of course that it is perfectly possible for $i \in \Fix \pi$ and also $i \in \Fix \rho$, that is, there may well be elements fixed by more than one of a pair of **disjoint permutations**.

## Also see

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 79$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $9$: Permutations: Definition $9.4$