Fitting height and lengths of laws in finite solvable groups

Let G be a finite solvable group, and let h(G) denote its Fitting height, namely the length of a shortest normal series in G with nilpotent factors. We show, that any law in G has length at least h(G). This result is then used to improve a previously given bound on the nonsolvable length of finite nonsolvable groups

Free subgroups of inverse limits of iterated wreath products of non-abelian finite simple groups in primitive actions

Abstract Let 𝒲 = { G i ∣ 1 ≤ i ∈ ℕ } {\mathcal{W}=\{G_{i}\mid 1\leq i\in\mathbb{N}\}} be a set of non-abelian finite simple groups. Set W 1 = G 1 {W_{1}=G_{1}} and choose a faithful transitive primitive W 1 W_{1} -set Δ 1 \varDelta_{1} . Assume that we have already constructed W n - 1 W_{n-1} and chosen a transitive faithful primitive W n - 1 W_{n-1} -set Δ n - 1 \varDelta_{n-1} . The group W n W_{n} is then defined as W n = G n ⁢ wr Δ n - 1 ⁡ W n - 1 {W_{n}=G_{n}\operatorname{wr}_{\varDelta_{n-1}}W_{n-1}} . If W is the inverse limit W = lim ← ⁡ ( W n , ρ n ) {W=}{\varprojlim(W_{n},\rho_{n})} with respect to the natural projections ρ n : W n → W n - 1 {\rho_{n}\colon W_{n}\to W_{n-1}} , we prove that, for each k ≥ 2 k\geq 2 , the set of k-tuples of W that freely generate a free subgroup of rank k is comeagre in W k W^{k} and its complement has Haar measure zero.</jats:p

Sur quelques aspects de la donation Jacques et Guy Thuillier

Le musée des beaux-arts de Nancy s’est enrichi en 1999 d’une exceptionnelle donation anonyme d’arts graphiques d’environ mille cinq cents œuvres, dessins et estampes. Cette collection, encore très confidentielle, contient des pièces rares, acquises par un œil exceptionnel. Composée pour la plus grande partie (plus de douze mille feuilles) d’estampes, art exigeant et peu connu, elle invite le spectateur attentif à la découverte de pièces d’une richesse insoupçonnée. Cet ensemble, fruit d’une v..