Permutations: Notation that uses composition with exponents?

Question

Notation question.

Wiki Compositions of permutations claims that compositions :

$$\sigma \bullet \pi$$

is the function that maps any element $x$ of the set to:

$$\sigma (\pi (x))$$

and that another notation for permutations is denoted by an exponent, where $\sigma$ acting on $x$ is denoted:

$$x^{\sigma}$$

then the above product is denoted by:

$$x^{\sigma \bullet \pi} = (x^{\sigma})^{\pi}$$

I'm under the impression that for example: $$\sigma = \{(x_0, x_1) (x_1, x_0)\}$$ $$\pi = \{(x_2, x_3) (x_3, x_2)\}$$

Can anyone further clarify the exponent notation with an example?

ANSWER

$$x^{\sigma + \pi} = (x^{\sigma})^{\pi} = (\sigma \circ \pi)(x) = \sigma(\pi(x)))$$

Answer 1

$x^\sigma$ is just another notation for $\sigma(x)$. They mean exactly the same thing.

However, the "best" meaning of $\sigma \circ \pi$ varies between the two notations.

If you write $\sigma(x)$, then you would like $$(\sigma \circ \pi)(x) = \sigma(\pi(x))$$ which means that when computing $\sigma \circ \pi$, you do $\pi$ first, then $\sigma$.

If you write $x^\sigma$, then you would like $$x^{\sigma \circ \pi} = (x^\sigma)^\pi$$ which means that when computing $\sigma \circ \pi$, you do $\sigma$ first, then $\pi$.

Mixing the two notations in the same context can lead to confusion and "terrible formulas" (as Serre put it in one of his books) like $x^{\sigma \circ \pi} = (x^\pi)^\sigma$. Try to avoid ever doing this.