Exercise 2, § 4

Let $n\geqslant 1$ and $p\in [1,+\infty ]$. Endow $\mathbf{R}^n$ with the norm

$n$

$\|x\|=(\sum_{i=1}|x_i|^p)^{1/p}$ if $p\not = +\infty$, and $\|x\|=$ sup$_i|x_i|$ if $p= +\infty$. Let G be the group of isometries of the metric space $\mathbf{R}^n$ endowed with this norm.

a) Let $\sigma$ be a permutation of $\{1, . . . , n\}$ and $\varepsilon \in \{-1,1\}^n$. Let $(e_i)$ be the canonical basis of $\mathbf{R}^n$. The unique endomorphism of $\mathbf{R}^n$ such that $e_i\mapsto \varepsilon_ie_{\sigma(i)}$ belongs to G. The set of isometries of this form is a subgroup W of G; it is an extension of $\mathfrak{S}_n$ by $(\mathbf{Z}/2\mathbf{Z})^n$.

b) The group G is compact.

c) The action of the group G on $\mathbf{R}^n$ is irreducible; the group G is contained in the orthogonal group of the quadratic form $\sum x^2_i$.

d) If $p\not = 2$, then G = W.