Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G, that is, $AP=PA$. Let $A_{sG}$ be the subgroup of A generated by all S-permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S-permutable subgroups of G containing A. We prove that if G is a soluble group, then S-permutability is a transitive relation in G if and only if the nilpotent residual $G^{\mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$, that is, $G^{\mathfrak {N}}\cap A^{sG}= G^{\mathfrak {N}}\cap A_{sG}$ for every subnormal subgroup A of G.

A well-known theorem of Philip Hall states that if a group G has a nilpotent normal subgroup N such that $G/N'$ is nilpotent, then G itself is nilpotent. We say that a group class 𝔛 is a Hall class if it contains every group G admitting a nilpotent normal subgroup N such that $G/N'$ belongs to 𝔛. Hall classes have been considered by several authors, such as Plotkin [‘Some properties of automorphisms of nilpotent groups’, Soviet Math. Dokl. 2 (1961), 471–474] and Robinson [‘A property of the lower central series of a group’, Math. Z. 107 (1968), 225–231]. A further detailed study of Hall classes is performed by us in another paper [‘Hall classes of groups’, to appear] and we also investigate the behaviour of the class of finite-by-𝔜 groups for a given Hall class 𝔜 [‘Hall classes in linear groups’, to appear]. The aim of this paper is to prove that for most natural choices of the Hall class 𝔜, also the classes $(\mathbf{L}\mathfrak{F})\mathfrak{Y}$ and 𝔅𝔜 are Hall classes, where L𝔉 is the class of locally finite groups and 𝔅 is the class of locally finite groups of finite exponent.

Gagola and Lewis [‘A character theoretic condition characterizing nilpotent groups’, Comm. Algebra 27 (1999), 1053–1056] proved that a finite group G is nilpotent if and only if $\chi (1)^{2}$ divides $\lvert G:\textrm {ker}\,\chi \rvert $ for every irreducible character $\chi $ of G. The theorem was later generalised by using monolithic characters. We generalise the theorem further considering only strongly monolithic characters. We also give some criteria for solvability and nilpotency of finite groups by their strongly monolithic characters.

We say that a group G of local (maybe formal) biholomorphisms satisfies the uniform intersection property if the intersection multiplicity $(\phi (V), W)$ takes only finitely many values as a function of G for any choice of analytic sets V and W of complementary dimension. In dimension $2$ we show that G satisfies the uniform intersection property if and only if it is finitely determined – that is, if there exists a natural number k such that different elements of G have different Taylor expansions of degree k at the origin. We also prove that G is finitely determined if and only if the action of G on the space of germs of analytic curves has discrete orbits.

We investigate unramified extensions of number fields with prescribed solvable Galois group G and certain extra conditions. In particular, we are interested in the minimal degree of a number field K, Galois over $\mathbb {Q}$ , such that K possesses an unramified G-extension. We improve the best known bounds for the degree of such number fields K for certain classes of solvable groups, in particular for nilpotent groups.

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

The class of all monolithic (that is, subdirectly irreducible) groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.

An isotopic to the identity map of the 2-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the 2-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular, we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the 2-torus has a global fixed point.

Let $G$ be a finite group and let $p$ be a prime factor of $|G|$. Suppose that $G$ is solvable and $P$ is a Sylow $p$-subgroup of $G$. In this note, we prove that $P{\vartriangleleft}G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all irreducible monomial $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

If $k$ is a positive integer, a group $G$ is said to have the $FE_{k}$-property if for each element $g$ of $G$ there exists a normal subgroup of finite index $X(g)$ such that the subgroup $\langle g,x\rangle$ is nilpotent of class at most $k$ for all $x\in X(g)$. Thus, $FE_{1}$-groups are precisely those groups with finite conjugacy classes ($FC$-groups) and the aim of this paper is to extend properties of $FC$-groups to the case of groups with the $FE_{k}$-property for $k>1$. The class of $FE_{k}$-groups contains the relevant subclass $FE_{k}^{\ast }$, consisting of all groups $G$ for which to every element $g$ there corresponds a normal subgroup of finite index $Y(g)$ such that $\langle g,U\rangle$ is nilpotent of class at most $k$, whenever $U$ is a nilpotent subgroup of class at most $k$ of $Y(g)$.

In this paper we study the algebraic structure of ω-stable bilinear maps, arbitrary rings, and nilpotent groups. We will also provide rather complete structure theorems for the above structures in the finite Morley rank case.

Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma \left( H \right)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma \left( H \right)$ if and only if $H\,=\,LK$. In this paper, we study the connectivity, diameter, clique number, and vertex chromatic number of $\Gamma \left( H \right)$. For instance, we show that if $\Gamma \left( H \right)$ has no isolated vertex, then $\Gamma \left( H \right)$ is connected with diameter at most 3. Also, we characterize all finitely groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma \left( H \right)$ is connected, and moreover, the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.

The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ in which all proper subgroups of cardinality $\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\aleph$ has cofinality strictly larger than $\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.

Let $G$ be a finite group. We denote by ${\it\nu}(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup and by $\text{Nil}_{G}(x)$ the set of elements $y\in G$ such that $\langle x,y\rangle$ is a nilpotent subgroup. A group $G$ is called an ${\mathcal{N}}$-group if $\text{Nil}_{G}(x)$ is a subgroup of $G$ for all $x\in G$. We prove that if $G$ is an ${\mathcal{N}}$-group with ${\it\nu}(G)>\frac{1}{12}$, then $G$ is soluble. Also, we classify semisimple ${\mathcal{N}}$-groups with ${\it\nu}(G)=\frac{1}{12}$.

We show that the space of actions of every finitely generated, nilpotent group by $C^1$ orientation-preserving diffeomorphisms of the circle is path-connected. This is done via a general result that allows any given action on the interval to be connected to the trivial one by a continuous path of topological conjugates.