Groups whose proper subgroups are metahamiltonian-by-finite

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

This is the third volume of a comprehensive and elementary treatment of finite p -group theory. Topics covered in this volume: impact of minimal nonabelian subgroups on the structure of p -groups, classification of groups all of whose nonnormal subgroups have the same order, degrees of irreducible characters of p -groups associated with finite algebras, groups covered by few proper subgroups, p -groups of element breadth 2 and subgroup breadth 1, exact number of subgroups of given order in a metacyclic p -group, soft subgroups, p -groups with a maximal elementary abelian subgroup of order p 2 , p -groups generated by certain minimal nonabelian subgroups, p -groups in which certain nonabelian subgroups are 2-generator. The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.

Let Γ be a nonelementary Kleinian group and H<ΓH<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).

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

§ 136 p-groups in which certain proper nonabelian subgroups are two-generator

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].

Algorithms for Matrix Groups and the Tits Alternative

Let Γ < SL n ( $$\mathbb{Z}$$ ) be a subgroup of finite index, where n ≥ 5. Suppose Γ acts continuously on a manifold M, where π1(M) = $$\mathbb{Z}$$ n , preserving a measure that is positive on open sets. Further assume that the induced Γ action on H 1(M) is non-trivial. We show there exists a finite index subgroup Γ′ < Γ and a Γ′ equivariant continuous map ψ : M → $$\mathbb{T}$$ n that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where π1(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with Γ actions.

On groups with uncountably many subgroups of finite index

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.

A profinite group is called strongly complete if every subgroup of finite index is open and of type (AF) if it has only finitely many subgroups of any fixed index. In this paper it is shown that a topologically finitely generated abelian by pro-nilpotent profinite group is strongly complete, and that a pro-solvable profinite group is strongly complete if is of type (AF).

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.

This note investigates torsion-free abelian groups G of finite rank which embed, as subgroups of finite index, in a finite direct sum C of subgroups of the additive group of rational numbers. Specifically, we examine the relationship between G and C when the index of G in C is minimal. Some properties of Warfield duality are developed and used (in the case that G is locally free) to relate our results to earlier ones by Burkhardt and Lady.

Affine structures on a class of virtually nilpotent groups