# Sign of Composition of Permutations

## Theorem

Let $n \in \N$ be a natural number.

Let $N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$.

Let $S_n$ denote the set of permutations on $N_n$.

Let $\map \sgn \pi$ denote the sign of $\pi$ of a permutation $\pi$ of $N_n$.

Let $\pi_1, \pi_2 \in S_n$.

Then:

- $\map \sgn {\pi_1} \map \sgn {\pi_2} = \map \sgn {\pi_1 \circ \pi_2}$

where $\pi_1 \circ \pi_2$ denotes the composite of $\pi_1$ and $\pi_2$.

## Proof

From Sign of Permutation on n Letters is Well-Defined, it is established that the sign each of $\pi_1$, $\pi_2$ and $\pi_1 \circ \pi_2$ is either $+1$ and $-1$.

By Existence and Uniqueness of Cycle Decomposition, each of $\pi_1$ and $\pi_2$ has a unique cycle decomposition.

Thus each of $\pi_1$ and $\pi_2$ can be expressed as the composite of $p_1$ and $p_2$ transpositions respectively.

Thus $\pi_1 \circ \pi_2$ can be expressed as the composite of $p_1 + p_2$ transpositions.

From Sum of Even Integers is Even, if $p_1$ and $p_2$ are both even then $p_1 + p_2$ is even.

In this case:

- $\map \sgn {\pi_1} = 1$
- $\map \sgn {\pi_2} = 1$
- $\map \sgn {\pi_1} \map \sgn {\pi_2} = 1 = 1 \times 1$

This needs considerable tedious hard slog to complete it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Although this article appears correct, it's inelegant. There has to be a better way of doing it.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

This article needs proofreading.Please check it for mathematical errors.If you believe there are none, please remove `{{Proofread}}` from the code.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Proofread}}` from the code. |

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 81$ - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory