We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor–Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

We show that there are only finitely many primes $p$ such that $\mathrm{PSL} (2, p)$ has a minimal generating set of size four.

Supplementary materials are available with this article.

In this paper we introduce the notion of finite virtual length for profinite groups (that is, every series has a bounded number of infinite factors) and we prove a Jordan–Hölder type theorem for profinite groups with finite virtual length. More structural results are provided in the pronilpotent and $p$-adic analytic cases.

Let $G$ be a finite $p$-solvable group and let ${G}^{\ast } $ be the set of elements of primary and biprimary orders of $G$. Suppose that the conjugacy class sizes of ${G}^{\ast } $ are $\{ 1, {p}^{a} , n, {p}^{a} n\} $, where the prime $p$ divides the positive integer $n$ and ${p}^{a} $ does not divide $n$. Then $G$ is, up to central factors, a $\{ p, q\} $-group with $p$ and $q$ two distinct primes. In particular, $G$ is solvable.

Many problems about local analysis in a finite group G reduce to a special case in which G has a large normal p-subgroup satisfying several restrictions. In 1983, R. Niles and G. Glauberman showed that every finite p-group S of nilpotence class at least 4 must have two characteristic subgroups S1 and S2 such that, whenever S is a Sylow p-subgroup of a group G as above, S1 or S2 is normal in G. In this paper, we prove a similar theorem with a more explicit choice of S1 and S2.

Let G be a finite p-solvable group. We describe the structure of the p-complements of G when the set of p-regular conjugacy classes has exactly three class sizes. For instance, when the set of p-regular class sizes of G is {1, pa, pam} or {1, m, pam} with (m, p) = 1, then we show that m = qb for some prime q and the structure of the p-complements of G is determined.

Let 𝕂 be a field, char(𝕂)≠2, and G a subgroup of GL(n,𝕂). Suppose g↦g♯ is a 𝕂-linear antiautomorphism of G, and then define G1={g∈G∣g♯g=I}. For C being the centraliser 𝒞G (G1) , or any subgroup of the centre 𝒵(G) , define G(C) ={g∈G∣g♯g∈C}. We show that G(C) is a subgroup of G, and study its structure. When C=𝒞G (G1) , we have that G(C) =𝒩G (G1) , the normaliser of G1 in G. Suppose 𝕂 is algebraically closed, 𝒞G (G1)consists of scalar matrices and G1 is a connected subgroup of an affine group G. Under the latter assumptions, 𝒩G (G1)is a self-normalising subgroup of G. This holds for a number of interesting pairs (G,G1); in particular, for those that we call parabolic pairs. As well, for a certain specific setting we generalise a standard result about centres of Borel subgroups.

c-Sections of maximal subgroups in a finite group and their relation to solvability have been extensively researched in recent years. A fundamental result due to Wang [‘C-normality of groups and its properties’, J. Algebra 180 (1998), 954–965] is that a finite group is solvable if and only if the c-sections of all its maximal subgroups are trivial. In this paper we prove that if for each maximal subgroup of a finite group G, the corresponding c-section order is smaller than the index of the maximal subgroup, then each composition factor of G is either cyclic or isomorphic to the O’Nan sporadic group (the converse does not hold). Furthermore, by a certain ‘refining’ of the latter theorem we obtain an equivalent condition for solvability. Finally, we provide an existence result for large subgroups in the sense of Lev [‘On large subgroups of finite groups’ J. Algebra 152 (1992), 434–438].

Let G be a finite group and let H≤G. We refer to |H||CG(H)| as the Chermak–Delgado measure ofH with respect to G. Originally described by Chermak and Delgado, the collection of all subgroups of G with maximal Chermak–Delgado measure, denoted 𝒞𝒟(G), is a sublattice of the lattice of all subgroups of G. In this paper we note that if H∈𝒞𝒟(G)then H is subnormal in G and prove that if K is a second finite group then 𝒞𝒟(G×K)=𝒞𝒟(G)×𝒞𝒟(K) . We additionally describe the 𝒞𝒟(G≀Cp)where G has a nontrivial centre and p is an odd prime and determine conditions for a wreath product to be a member of its own Chermak–Delgado lattice. We also examine the behaviour of centrally large subgroups, a subset of the Chermak–Delgado lattice.

The normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth ndim (G).

We consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed. We obtain a full description in the case in which for each prime divisor p of the order of the group there exists a noncentral conjugacy class of p-power size.

In Cossey and Stonehewer [‘On the rarity of quasinormal subgroups’, Rend. Semin. Mat. Univ. Padova125 (2011), 81–105] it is shown that for any odd prime p and integer n≥3, there is a finite p-group G of exponent pn containing a quasinormal subgroup H of exponent pn−1 such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, pn−1 or, when n≥4 , pn−2. Thus large sections of these groups are devoid of quasinormal subgroups. The authors ask in that paper if there is a nontrivial subgroup-theoretic property 𝔛 of finite p-groups such that (i) 𝔛 is invariant under subgroup lattice isomorphisms and (ii) every chain of 𝔛-subgroups of a finite p-group can be refined to a composition series of 𝔛-subgroups. Failing this, can such a chain always be refined to a series of 𝔛-subgroups in which the intervals between adjacent terms are restricted in some significant way? The present work embarks upon this quest.

Assume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to Cp∞. We show that if G is nilpotent, then the pro-p completion map induces an isomorphism for any discrete -module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map is an isomorphism for any discrete -module M of finite p-power order. Moreover, if G lacks any Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

Groups having exactly one normaliser are well known. They are the Dedekind groups. All finite groups having exactly two normalisers were classified by Pérez-Ramos [‘Groups with two normalizers’, Arch. Math.50 (1988), 199–203], and Camp-Mora [‘Locally finite groups with two normalizers’, Comm. Algebra28 (2000), 5475–5480] generalised that result to locally finite groups. Then Tota [‘Groups with a finite number of normalizer subgroups’, Comm. Algebra32 (2004), 4667–4674] investigated properties (such as solubility) of arbitrary groups with two, three and four normalisers. In this paper we prove that every finite group with at most 20 normalisers is soluble. Also we characterise all nonabelian simple (not necessarily finite) groups with at most 57 normalisers.

In the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.

We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL2(ℝ). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL2(ℝ) on these spaces, and give an application to the study of the Cremona group.

Let G be a finite p-solvable group. We prove that if G has exactly two conjugacy class sizes of p′-elements of prime power order, say 1 and m, then m=paqb, for two distinct primes p and q, and G either has an abelian p-complement or G=PQ×A, with P and Q a Sylow p-subgroup and a Sylow q-subgroup of G, respectively, and A is abelian. In particular, we provide a new extension of Itô’s theorem on groups having exactly two class sizes for elements of prime power order.

If G is a semisimple Lie group of real rank at least two and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image.

Let G be a group of odd order that contains a non-central element x whose order is either a prime p ≥ 5 or 3l, with l ≥ 2. Then, in , the group of units of ℤG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large integers m we have that 〈um, vm〉 = 〈um〉 ∗ 〈vm〉 ≌ ℤ ∗ ℤ

The following theorem is proved. Let m, k and n be positive integers. There exists a number η=η(m,k,n) depending only on m, k and n such that if G is any residually finite group satisfying the condition that the product of any η commutators of the form [xm,y1,…,yk ]is of order dividing n, then the verbal subgroup of G corresponding to the word w=[xm,y1,…,yk ]is locally finite.