Let ${𝔖}_n$ denote the symmetric group with $n$ letters, and $g\left(n\right)$ the maximal order of an element of ${𝔖}_n$. If the standard factorization of $M$ into primes is $M=q_1^{{\alpha }_1}{q}_2^{{\alpha }_2}...q_k^{{\alpha }_k}$, we define $\ell \left(M\right)$ to be $q_1^{{\alpha }_1}+q_2^{{\alpha }_2}+...+q_k^{{\alpha }_k}$; one century ago, E. Landau proved that $g\left(n\right)={max}_{\ell \left(M\right)\le n}M$ and that, when $n$ goes to infinity, $logg\left(n\right)\sim \sqrt{nlog\left(n\right)}$.There exists a basic algorithm to compute $g\left(n\right)$ for $1\le n\le N$; its running time is $𝒪\left(N^{3/2}/\sqrt{logN}\right)$ and the needed memory is $𝒪\left(N\right)$; it allows computing $g\left(n\right)$ up to, say, one million. We describe an algorithm to calculate $g\left(n\right)$ for $n$ up to ${10}^{15}$. The main idea is to use the so-called $\ell$-superchampion...

We consider the following notion of largeness for subgroups of $S_{\infty }$. A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{\infty }$ can be extended to a large free subgroup of $S_{\infty }$, and, under Martin’s Axiom, any free subgroup of $S_{\infty }$ of cardinality less than can also be extended to a large free subgroup of $S_{\infty }$. Finally, if Gₙ are countable groups, then...

Deux codages sont utilisés sur l’ensemble des permutations ou ordres totaux sur un ensemble fini à $n$ éléments et à chacun de ces codages est associé un produit direct d’ordres totaux. On démontre que le diagramme du treillis permutoèdre (ou ordre de Bruhat faible sur le groupe symétrique $S_n$) est intersection des diagrammes des deux produits directs de $n-1$ ordres totaux à $2,3,...,n$ éléments.