# What is a permutation of a Group?

I've become confused with a certain sentence of an abstract algebra text.

The author is talking about the Field extension $ℚ(x_1, x_2, ... , x_n)/ℚ$, where each $x_i$ is one of the roots of a polynomial of degree $n$.

He follows to explain that "the most important property of $ℚ(x_1, x_2, ... , x_n)$ is that it is symmetric with respect to $x_1, x_2, ... , x_n$, in the sense that any permutation $*$ of $x_1, x_2, ... , x_n$ extends to a bijection * of $ℚ(x_1, x_2, ... , x_n)$ defined by

$$* f(x_1, ... , x_n) = f(*x_1, ... , *x_n)$$

for each rational function $f$ of $x_1, ... , x_n$. Moreover, this bijection * obviously satisfies

$$*(f + g) = *f + *g,$$ $$*(fg) = (*f)(*g)$$

The issue is that I'm not really understanding what this permutation $*$ is, nor why it has the properties that the author claims it has.

I also don't get what he means by "symmetric". He was talking before about how each coefficient $a_i$ of the polynomial is a function of the roots. He called them "elementary symmetric functions". I guess -and might be completely wrong here- that the name has something to do with the fact that substituting a certain root $x_j$ with $x_k$ in any of these 'symmetric functions' wouldn't mess up the equation.

I don't know much about Group theory, so forgive me if I ask too many questions about the notation/terminology of the replies

I would really appreciate any help/thoughts!

## 1 Answer

The title of your question contains the expression “permutation of a Group” and you state that you “don't know much about Group theory”. However, your questions have very little to do with group theory.

What the author states is that for every permutation $\sigma$ of the set $\{x_1,x_2,\ldots,x_n\}$, there is a bijection of $\mathbb{Q}(x_1,x_2,\ldots,x_n)$ into itself thus defined: every rational function $f\in\mathbb{Q}(x_1,x_2,\ldots,x_n)$ is mapped int $\sigma f\in\mathbb{Q}(x_1,x_2,\ldots,x_n)$, where $\sigma f(x_1,x_2,\ldots,x_n)=f\bigl(\sigma(x_1),\sigma(x_2),\ldots,\sigma(x_n)\bigr)$.

The symmetric elementary polynomials are\begin{align}\sigma_1(x_1,x_2,\ldots,x_n)&=\sum_{k=1}^nx_k\\\sigma_2(x_1,x_2,\ldots,x_n)&=\sum_{i<j}x_ix_j\\&\cdots\\\sigma_n(x_1,x_2,\ldots,x_n)&=x_1x_2\ldots x_n\end{align}and if $x_1,x_2,\ldots,x_n$ are the roots of a monic polynomial $P(x)$, then$$P(x)=x^n-\sigma_1(x_1,x_2,\ldots,x_n)x^{n-1}+\sigma_2(x_1,x_2,\ldots,x_n)x^{n-2}-\cdots\pm\sigma_n(x_1,x_2,\ldots,x_n).$$