Let $\sigma$ be a permutation in $S_5$ with a sign of $-1$. Let $\pi$ be any other arbitrary permutation of $\{1,...,5\}$. What is the sign of the composition $\pi^{-1} \sigma \pi$?

Is there any particular theorem that applies to this, or is there any relationship between the signs of permutations of the same dimension? Any help would be appreciated! Thank you.

Edit: The composition of $\pi$ and $\pi^{-1}$ is the identity, and we know that the sign of the identity is $1$, by definition. It follows that: $$sgn(\pi^{-1}\sigma\pi)=sgn(\pi^{-1})sgn(\sigma)sgn(\pi)=-1\cdot sgn(\pi^{-1})sgn(\pi)=-1\cdot sgn(\pi^{-1}\pi)=-1\cdot 1=-1$$


The relevant theorem is this: the sign of a product is the product of the signs. Using this, we can see immediately that $\pi^{-1} \sigma \pi$ and $\sigma$ have the same sign.


The invariant definition of the signature of the permutation of a finite set, one that does not depend on the labeling of the elements, but only on the cycle structure of the permutation is

$$\text{sign}(\sigma) = (-1)^{ \text{ # cycles of $\sigma$ of even length}}$$

So we have the Triple E rule : a permutation is Even $\iff$ Even number of Even cycles.

Now the permutations $\pi^{-1} \sigma \pi$ and $\sigma$ have the same cycle structure so their signatures are the same.


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