Orazio Puglisi

Positive Definite Functions of Diagonal Limits of Finite Alternating Groups

The normalized positive definite class functions are determined for all those direct limits of finite alternating groups for which the embeddings are natural in the sense that every non-trivial -orbit in is natural.

Serial subalgebras of finitary Lie algebras

A Lie subalgebra L of glK(V ) is said to be finitary if it consists of elements of finite rank. We show that, if L acts irreducibly on V , and if V is infinite-dimensional, then every non-trivial ascendant Lie subalgebra of L acts irreducibly on V too. When CharK 6= 2, it follows that the locally solvable radical of such L is trivial. In general, locally solvable finitary Lie algebras over fields of characteristic 6= 2 are hyperabelian.

Free products of finitary linear groups

In this note we show that the free product of any family of groups which are finitary linear over fields of the same characteristic p ≥ 0, is still finitary linear over a field of characteristic p.

Periodic Groups Covered by Transitive Subgroups of Finitary Permutations or by Irreducible Subgroups of Finitary Transformations

Let X be either the class of all transitive groups of finitary permutations, or the class of all periodic irreducible finitary linear groups. We show that almost primitive X-groups are countably recognizable, while totally imprimitive X-groups are in general not countably recognizable. In addition we derive a structure theorem for groups all of whose countable subsets are contained in totally imprimitive X-subgroups. It turns out that totally imprimitive p-groups in the class X are countably recognizable.

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.

Unipotent automorphisms of solvable groups

Abstract Let G be a solvable group and H a solvable subgroup of Aut ⁡ ( G )

Sylow subgroups of finitary linear groups

\operatorname{Aut}(G)

Locally Solvable Finitary Linear Groups

whose elements are n-unipotent. When H is finitely generated, we show that it stabilizes a finite series in G and conclude that H is nilpotent. If G furthermore has a characteristic series with torsion-free factors. the same conclusion as above holds without the extra assumption that H is finitely generated.

Unipotent finitary linear groups

AbstractIn this paper we describe the structure and the conjugacy classes of Sylow p-subgroups of FGL(V,