Coset posets of infinite groups

On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups

Decision problems and profinite completions of groups

A countable group is ℵ0-categorical if it is characterized up to isomorphism among all countable groups by its first order theory. We investigate countable ℵ0-categorical groups which are characteristically simple and have proper subgroups of finite index. We determine such groups up to isomorphism (Corollary 4 to Proposition 3), and we show that their theories are finitely axiomatizable (Proposition 2).Let K be a finite nonabelian simple group. In §3 we construct from K two denumberable groups, K* and K#, and use them to classify countable direct limits of finite Cartesian powers of K (Proposition 1). The results of §3 are used in §4 to show that K* and K# are ℵ0-categorical and to find axioms for their theories. By Corollary 2 to Proposition 2, K* and K# are counterexamples to the conjecture of U. Feigner [3, p. 309] that ℵ0-categorical groups are FC-solvable. Also in §4 the results mentioned in the first paragraph are proved. §2 is devoted to preliminary lemmas about finite Cartesian powers of K.A characteristic subgroup of a group G is a subgroup which is mapped onto itself by all automorphisms of G. G is characteristically simple if its only characteristic subgroups are itself and the identity subgroup. Each characteristic subgroup of a group is a union of orbits of the automorphism group of the group. If G is ℵ0-categorical, then as the automorphism group of G has only a finite number of orbits, G has only a finite number of characteristic subgroups.

On groups with uncountably many subgroups of finite index

A group is called metahamiltonian if all its non-abelian subgroups are normal. It is known that any infinite locally graded group whose proper subgroups are metahamiltonian is likewise metahamiltonian, and the aim of this paper is to describe the structure of locally graded groups whose proper subgroups contain a metahamiltonian subgroup of finite index.

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size [Formula: see text] subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

We show that if all proper subgroups of a locally graded group G are finite-by-abelian-by-finite, then G contains a finite normal subgroup N such that all proper subgroups of G/N are abelian-by-finite. Then we apply this result to the study of groups which are minimal-non- $${\mathcal {P}}$$ also for related group properties $${\mathcal {P}}$$ . Finally we see how for locally (soluble-by-finite) groups of infinite rank, it is enough to restrict attention to the proper subgroups with infinite rank.

Generalizing results in [1], [3], [5], and [7] it is shown that the algebra generated by the quasi-regular representation λ of a discrete groupG onl2 (G/H) satisfies a standard polynomial identity if and only if λ (G) contains an abelian normal subgroup of finite index. An analogon for topological groups is also proved.

We shall often need to consider the element Z =X Y, which is a parabolic transformation. Any parabolic transformation in r is conjugate to a power of Z. Let N be a normal subgroup of finite index ,u in r. The level n of N is defined as the least positive integer such that ZnCN. The conjugacy class of Z in r splits up into a finite number of equivalence classes under conjugacy by N. The number t of equivalence classes is called the parabolic class number of N. It is known that the integers ,u, n, t satisfy the relation

Let \(\mathcal{N}\) be a nonempty collection of normal subgroups of finite index of a group G and assume that \(\mathcal{N}\) is filtered from below, i.e., \(\mathcal{N}\) satisfies the following condition: $$\mbox{whenever }N_1,N_2\in \mathcal{N},\mbox{ there exists }N\in \mathcal{N}\mbox{ such that }N\le N_1\cap N_2.$$ Then one can make G into a topological group by considering \(\mathcal{N}\) as a fundamental system of neighborhoods of the identity element 1 of G. We refer to the corresponding topology on G as a profinite topology. If every quotient G/N\((N\in \mathcal{N})\) belongs to a certain class \(\mathcal{C}\), we say more specifically that the topology above is a pro - \(\mathcal{C}\)topology.

In this paper, we shall deal with periodic groups, in which each element has a prime power order. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each [Formula: see text]-element of [Formula: see text] is of order [Formula: see text]. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each finite [Formula: see text]-subgroup of [Formula: see text] is of order [Formula: see text]. Here, [Formula: see text] denotes the set of all primes dividing the order of some element of [Formula: see text]. Our main results are the following four theorems. Theorem 1: Let [Formula: see text] be a finitely generated [Formula: see text]-group. Then [Formula: see text] has only a finite number of normal subgroups of finite index. Theorem 4: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a locally finite group. Theorem 7: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a finite group. Theorem 8: Let [Formula: see text] be a [Formula: see text]-group satisfying [Formula: see text]. Then [Formula: see text] is a locally finite group.

Profinite completions and isomorphic finite quotients

The uniqueness of the translation-invariant extreme Gibbs measure for the antiferromagnetic Potts model with an external field and the existence of an uncountable number of extreme Gibbs measures for the Ising model with an external field on the Cayley tree are proved. The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. The periodic extreme Gibbs measures, which are invariant with respect to subgroups of index 2, are constructed for the Ising model with zero external field. From these measures, the existence of an uncountable number of nonperiodic extreme Gibbs measures for the antiferromagnatic Ising model follows.

A proper subgroup H of a group G is said to be strongly isolated if it contains the centralizer of any nonidentity element of H and 2-isolated if the conditions >CG(g) ∩ H ≠ 1 and 2∈π(CG(g)) imply that CG(g)≤H. An involution i in a group G is said to be finite if |iig| < ∞ (for any g∈ G). In the paper we study a group G with finite involution i and with a 2-isolated locally finite subgroup H containing an involution. It is proved that at least one of the following assertions holds:

Let Γ be a nonelementary Kleinian group and H&lt;ΓH&lt;Γ be a finitely generated, proper subgroup. We prove that if Γ has finite covolume, then the profinite completions of H and Γ are not isomorphic. If H has finite index in Γ, then there is a finite group onto which H maps but Γ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circle of ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, for example, limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic 3-manifold Vol(3)Vol(3) and that of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in PSL(2,C)PSL(2,C) is profinitely rigid, then so is its normalizer in PSL(2,C)PSL(2,C).