To the question of existence of exact irreducible representations of soluble groups of finite rank over locally finite field

We find characteristic subgroup of soluble torsion-free group of finite rank, whose structure determines sufficient conditions of existence of exact irreducible representations of the group over locally finite field.

An extension of the Glauberman ZJ-theorem

Let [Formula: see text] be an odd prime and let [Formula: see text], [Formula: see text] and [Formula: see text] denote the three different versions of Thompson subgroups for a [Formula: see text]-group [Formula: see text]. In this paper, we first prove an extension of Glauberman’s replacement theorem [G. Glauberman, A characteristic subgroup of a p-stable group, Canad. J. Math. 20 (1968) 1101–1135, Theorem 4.1]. Second, we prove the following: Let [Formula: see text] be a [Formula: see text]-stable group and [Formula: see text]. Suppose that [Formula: see text]. If [Formula: see text] is a strongly closed subgroup in [Formula: see text], then [Formula: see text], [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text]. Third, we show the following: Let [Formula: see text] be a [Formula: see text]-free group and [Formula: see text]. If [Formula: see text] is a strongly closed subgroup in [Formula: see text], then the normalizers of the subgroups [Formula: see text], [Formula: see text] and [Formula: see text] control strong [Formula: see text]-fusion in [Formula: see text]. We also prove a similar result for a [Formula: see text]-stable and [Formula: see text]-constrained group. Finally, we give a [Formula: see text]-nilpotency criteria, which is an extension of Glauberman–Thompson [Formula: see text]-nilpotency theorem.

Large normal subgroup growth and large characteristic subgroup growth

The fastest normal subgroup growth type of a finitely generated group is {n^{\log n}}. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let Γ be a group and Δ a subgroup of finite index. Suppose Δ has normal subgroup growth of type {n^{\log n}}. Does Γ have normal subgroup growth of type {n^{\log n}}? We give a positive answer in some cases, generalizing a result of Müller and the second author and a result of Gerdau. For instance, suppose G is a profinite group and H an open subgroup of G. We show that if H is a generalized Golod–Shafarevich group, then G has normal subgroup growth of type {n^{\log n}}. We also use our methods to show that one can find a group with characteristic subgroup growth of type {n^{\log n}}.

Characteristic subgroup lattices and Hopf–Galois structures

The Hopf–Galois structures on normal field extensions [Formula: see text] with [Formula: see text] are in one-to-one correspondence with the set of regular subgroups [Formula: see text] of [Formula: see text], the group of permutations of [Formula: see text] as a set, that are normalized by the left regular representation [Formula: see text]. Each such [Formula: see text] corresponds to a Hopf algebra [Formula: see text] that acts on [Formula: see text]. Such regular subgroups need not be isomorphic to [Formula: see text] but must have the same order. One can divide all such [Formula: see text] into collections [Formula: see text], where [Formula: see text] is the set of those regular [Formula: see text] normalized by [Formula: see text] and isomorphic to a given abstract group [Formula: see text], where [Formula: see text]. There exists an injective correspondence between the characteristic subgroups of a given [Formula: see text] and the set of subgroups of [Formula: see text] stemming from the Galois correspondence between sub-Hopf algebras of [Formula: see text] and intermediate fields [Formula: see text], where [Formula: see text]. We utilize this correspondence to show that for certain pairs [Formula: see text], the collection [Formula: see text] must be empty. This also shows that for these [Formula: see text], there do not exist skew braces with additive group isomorphic to [Formula: see text] and circle group isomorphic to [Formula: see text].

A Note on the Square Subgroups of Decomposable Torsion-Free Abelian Groups of Rank Three

A hypothesis stated in [16] is confirmed for the case of associative rings. The answers to some questions posed in the mentioned paper are also given. The square subgroup of a completely decomposable torsion-free abelian group is described (in both cases of associative and general rings). It is shown that for any such a group A, the quotient group modulo the square subgroup of A is a nil-group. Some results listed in [16] are generalized and corrected. Moreover, it is proved that for a given abelian group A, the square subgroup of A considered in the class of associative rings, is a characteristic subgroup of A.

The metanorm, a characteristic subgroup: Embedding properties

Thenormof a group was introduced by R. Baer as the intersection of all normalizers of subgroups, and it was later proved that the norm is always contained in the second term of the upper central series of the group. The aim of this paper is to study embedding properties of themetanormof a group, defined as the intersection of all normalizers of non-abelian subgroups. The metanorm is related to the so-calledmetahamiltonian groups, i.e. groups in which all non-abelian subgroups are normal, and it is known that every locally graded metahamiltonian group is finite over its second centre. Among other results, it is proved here that ifGis a locally graded group whose metanormMis not nilpotent, then{M^{\prime}/M^{\prime\prime}}is a small eccentric chief factor and it is the only obstruction to a strong hypercentral embedding ofMinG.

On the structure of virtually nilpotent compact p-adic analytic groups

LetGbe a compactp-adic analytic group. We recall the well-understood finite radical{\Delta^{+}}and FC-centre Δ, and introduce ap-adic analogue of Roseblade’s subgroup{\mathrm{nio}(G)}, the unique largest orbitally sound open normal subgroup ofG. Further, whenGis nilpotent-by-finite, we introduce the finite-by-(nilpotentp-valuable) radical{\mathbf{FN}_{p}(G)}, an open characteristic subgroup ofGcontained in{\mathrm{nio}(G)}. By relating the already well-known theory of isolators with Lazard’s notion ofp-saturations, we introduce the isolated lower central (resp. isolated derived) series of a nilpotent (resp. soluble)p-valuable group, and use this to study the conjugation action of{\mathrm{nio}(G)}on{\mathbf{FN}_{p}(G)}. We emerge with a structure theorem forG,1\leq\Delta^{+}\leq\Delta\leq\mathbf{FN}_{p}(G)\leq\mathrm{nio}(G)\leq G,in which the various quotients of this series of groups are well understood. This sheds light on the ideal structure of the Iwasawa algebras (i.e. the completed group ringskG) of such groups, and will be used in future work to study the prime ideals of these rings.

Three-dimensional trajectories and network analyses of group behaviour within chimney swift flocks during approaches to the roost

Chimney swifts ( Chaetura pelagica ) are highly manoeuvrable birds notable for roosting overnight in chimneys, in groups of hundreds or thousands of birds, before and during their autumn migration. At dusk, birds gather in large numbers from surrounding areas near a roost site. The whole flock then employs an orderly, but dynamic, circling approach pattern before rapidly entering a small aperture en masse . We recorded the three-dimensional trajectories of ≈1 800 individual birds during a 30 min period encompassing flock formation, circling, and landing, and used these trajectories to test several hypotheses relating to flock or group behaviour. Specifically, we investigated whether the swifts use local interaction rules based on topological distance (e.g. the n nearest neighbours, regardless of their distance) rather than physical distance (e.g. neighbours within x m, regardless of number) to guide interactions, whether the chimney entry zone is more or less cooperative than the surrounding flock, and whether the characteristic subgroup size is constant or varies with flock density. We found that the swift flock is structured around local rules based on physical distance, that subgroup size increases with density, and that there exist regions of the flock that are less cooperative than others, in particular the chimney entry zone.