Felix Leinen

A reduction theorem for perfect locally finite minimal non-FC groups

A group G is said to be a minimal non-FC group , if G contains an infinite conjugacy class, while every proper subgroup of G merely has finite conjugacy classes. The structure of imperfect minimal non- FC groups is quite well-understood. These groups are in particular locally finite. At the other end of the spectrum, a perfect locally finite minimal non- FC group must be a p -group. And it has been an open question for quite a while now, whether such groups exist or not.

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.

Existentially Closed Groups in Specific Classes

This survey article is intended to make the reader familiar with the algebraic structure of existentially closed groups in specific group classes, and with the ideas and methods involved in this area of group theory. We shall try to give a fairly complete account of the theory, but there will be a certain emphasis on classes of nilpotent groups, locally finite groups, and extensions.

Local systems in simple 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 every simple finitary Lie algebra over a field of characteristic ≠2, 3, 5, 7 has a local system consisting of perfect central extensions of finite-dimensional simple Lie algebras.

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.

Countable Closed LFC-Groups with p-Torsion

Let LFC be the class of all locally FC-groups. We study the existentially closed groups in the class LFC p of all LFC-groups H whose torsion subgroup T(H) is a p-group. Differently from the situation in LFC, every existentially closed LFC p -group is already closed in LFC p , and there exist 2ℵ 0 countable closed LFCp-groups G. However, in the countable case, T(G) is up to isomorphism always a unique locally finite p-group with similar properties as the unique countable existentially closed locally finite p-group E p

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.