<Emphasis Type="Italic">C</Emphasis>*-simplicity of <Emphasis Type="Italic">n</Emphasis>-periodic products

The C*-simplicity of n-periodic products is proved for a large class of groups. In particular, the n-periodic products of any finite or cyclic groups (including the free Burnside groups) are C*-simple. Continuum-many nonisomorphic 3-generated nonsimple C*-simple groups are constructed in each of which the identity xⁿ = 1 holds, where n ≥ 1003 is any odd number. The problem of the existence of C*-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.     

A family of non-isomorphism results

We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV. For a given \({n\in N\cup\left\{\omega\right\}}\) , these groups of germs are free abelian groups of rank r, for r ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin’s theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associated groups of germs. Thus, if m ≠ n the groups mV and nV cannot be isomorphic. This answers a question of Brin.     

Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FP_n over a profinite ring R, analogous to the Bieri–Eckmann criteria for abstract groups. We use these to prove that the class of groups of type FP_n is closed under extensions, quotients by subgroups of type FP_n, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP_∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FP_n but not of type FP_n+1 over Z_? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FP_n fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.     

Verbal width in anabelian groups

The class A of anabelian groups is defined as the collection of finite groups without abelian composition factors. We prove that the commutator word [x₁, x₂] and the power word x₁^p have bounded width in A when p is an odd integer. By contrast, the word x³⁰ does not have bounded width in A. On the other hand, any given word w has bounded width for those groups G ∈ A whose composition factors are sufficiently large as a function of w. In the course of the proof we establish that sufficiently large almost simple groups cannot satisfy w as a coset identity.     

Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups

We consider a representation of quasi-endomorphisms of Abelian torsion-free groups of rank 4 bymatrices of order 4 over the field of rational numbers Q. We obtain a classification for quasi-endomorphism rings of Abelian torsion-free groups of rank 4 quasi-decomposable into a direct sum of groups A ₁, A ₂ of rank 1 and strongly indecomposable group B of rank 2 such that quasi-homomorphism groups Q ? Hom(A _i , B) and Q ? Hom(B, A _i ) for any i = 1, 2 have rank 1 or are zero. Moreover, for algebras from the classification we present necessary and sufficient conditions for their realization as quasi-endomorphism rings of these groups.     

Test elements in pro-<Emphasis Type="Italic">p</Emphasis> groups with applications in discrete groups

Let G be a group. An element g ∈ G is called a test element of G if for every endomorphism ? : G → G, ?(g) = g implies ? is an automorphism. We prove that for a finitely generated profinite group G, g ∈ G is a test element of G if and only if it is not contained in a proper retract of G. Using this result we prove that an endomorphism of a free pro-p group of finite rank which preserves an automorphic orbit of a nontrivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-p groups and Demushkin groups. By relating test elements in finitely generated residually finite-p Turner groups to test elements in their pro-p completions, we provide new examples of test elements in free discrete groups and surface groups. Moreover, we prove that the set of test elements of a free discrete group of finite rank is dense in the profinite topology.     

Characterization of ℝ-factorizable <Emphasis Type="Italic">G</Emphasis>-spaces

The ?-factorizability of G-spaces is characterized in the paper and the equivalence of ?-factorizability and ω-U property is proved for G-spaces with d-open actions of ω-narrow groups. It is shown that the ?-factorizability characterizes those compact coset spaces which are coset spaces of ω-narrow groups. The concepts of m- and M-factorizable G-spaces are introduced, which generalizes the corresponding concepts for topological groups.     

Fitting cores and supersolvable groups

Let A be a group. What can be said about the group B to ensure that A and the normal product AB belong to the same prescribed class of groups? Results in this direction are given for the classes of supersolvable groups, absolutely solvable groups and Lagrange groups.     

Finite tangled groups

We study the so-called finite tangled groups. These are the groups whose every subset containing 1 and closed under the operation x ○ y = xy ^?1 x is a subgroup. The general problem of studying such groups reduces to the case of tangled groups of odd order. We classify all finite nilpotent tangled groups.     

Periodic Groups Saturated with the Linear Groups of Degree 2 and the Unitary Groups of Degree 3 over Finite Fields of Odd Characteristic

Let M denote the set of the simple 3-dimensional unitary groups U₃ and the simple linear groups L₂ over finite fields of odd characteristic.We prove that each periodic group saturated with groups in M is locally finite and isomorphic to either U₃(Q) or L₂(Q) for a suitable locally finite field Q of odd characteristic.     

On <Emphasis Type="Italic">MP</Emphasis>-closed saturated formations of finite groups

A class of groups F is called MP-closed, if it contains every group G = AB such that the F-subgroup A permutes with every subgroup of B and the F-subgroup B permutes with every subgroup of A. We prove that the formation F that contains the class of all supersoluble groups is MP-closed if and only if the formation F(p) is MP-closed for any prime number p, where F is a maximal inner local screen of F. In particular, we prove that the formation of all groups with supersoluble Schmidt subgroups is MP-closed.     

The Groups of Motions of Some Three-Dimensional Maximal Mobility Geometries

We find the groups of motions of eight three-dimensional maximal mobility geometries. These groups are actions of just three Lie groups SL₂(R)ΔN, SL₂(C) _R , and SL₂(R)?SL₂(R) on the space R³, where N is a normal abelian subgroup. We also find explicit expressions for these actions.     

Conservative groups,indicability, and a conjecture of Howie

The class of locally p-indicable groups coincides with the class of $\text{Z}$_p conservative groups. Howie's conjecture is valid for locally p-indicable groups.     

Characterization of the alternating groups by their order and one conjugacy class length

Let G be a finite group, and let N(G) be the set of conjugacy class sizes of G. By Thompson’s conjecture, if L is a finite non-abelian simple group, G is a finite group with a trivial center, and N(G) = N(L), then L and G are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In this article, we investigate validity of Thompson’s conjecture under a weak condition for the alternating groups of degrees p+1 and p+2, where p is a prime number. This work implies that Thompson’s conjecture holds for the alternating groups of degree p + 1 and p + 2.     

Glider Representations of Group Algebra Filtrations of Nilpotent Groups

We continue the study of glider representations of finite groups G with given structure chain of subgroups e ? G ₁ ?… ? G _d = G. We give a characterization of irreducible gliders of essential length e ≤ d which in the case of p-groups allows to prove some results about classical representation theory. The paper also contains an introduction to generalized character theory for glider representations and an extension of the decomposition groups in the Clifford theory. Furthermore, we study irreducible glider representations for products of groups and nilpotent groups.     

Canonical and existential groups in universal classes of Abelian groups

Universal classes of Abelian groups are classified in terms of sets of finitely generated groups closed with respect to the discrimination operator. The notions of a principal universal class and a canonical group for such a class are introduced. For any universal class K, the class K^ec of existentially closed groups generated by the universal theory of K is described. It is proved that K^ec is axiomatizable and, therefore, the universal theory of K has a model companion.     

On the Malle conjecture and the self-twisted cover

For a large class of finite groups G, the number of Galois extensions E/Q of group G and discriminant |d _E | ≤ y is shown to grow at least like a power of y, for some specified positive exponent. The groups G are the regular Galois groups over Q and the counted extensions E/Q are obtained by specializing a given regular Galois extension F/Q(T). The extensions E/Q can further be prescribed any unramified local behavior at each suitably large prime p ≤ log(y)/δ for some δ ≥ 1. This result is a step toward the Malle conjecture on the number of Galois extensions of given group and bounded discriminant. The local conditions further make it a notable constraint on regular Galois groups over Q. The method uses a new version of Hilbert’s irreducibility theorem that counts the specialized extensions and not just the specialization points. A main tool for it is the self-twisted cover that we introduce.