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A group $G$ is said to have dense ${\cal CD}$-subgroups if each non-empty open interval of the subgroup lattice $L(G)$ contains a subgroup in the Chermak--Delgado lattice ${\cal CD}(G)$. In this note, we study finite groups satisfying this property.

We discuss examples of linear representations of finite groups as subgroups of the Riordan group. In particular, we show that the symmetric group of degree three has no faithful representation as a subgroup of the Riordan group over the complex numbers, but can be embedded as a subgroup of the Riordan group over a field of characteristic three.

There has been interest recently concerning when a left ordered group is locally indicable. Bergman and Tararin have shown that not all left ordered groups are locally indicable, but all known examples contain a nonabelian free subgroup. We shall show for a large class of groups not containing a nonabelian free subgroup, that any left ordered group in this class is locally indicable. Specifically this class is the smallest class of groups containing NS and Thompson's group of piecewise linear homeomorphisms of the unit interval, and is closed under taking subgroups, quotient groups, extensions and directed unions; here NS is the class of groups which do not contain a nonabelian subsemigroup. We shall also show that certain free products with an amalgamated cyclic subgroup are left orderable.

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $Π$-property in $G$ if there exists a chief series $\mathitΓ_G:1=G_0<G_1<\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\leq i\leq n$) of $\mathitΓ_G$, $|G/G_{i-1}:N_{G/G_{i-1}}(HG_{i-1}/G_{i-1}\cap G_i/G_{i-1})|$ is a $π(HG_{i-1}/G_{i-1}\cap G_i/G_{i-1})$-number. Our main results are listed here: Theorem A. Let $\mathfrak{F}$ be a solubly saturated formation containing $\mathfrak{U}$ and $E$ a normal subgroup of $G$ with $G/E\in \mathfrak{F}$. Let $X\unlhd G$ such that $F_p^*(E)\leq X\leq E$. Suppose that for any Sylow $p$-subgroup $P$ of $X$, every maximal subgroup of $P$ satisfies partial $Π$-property in $G$. Then one of the following holds: (1) $G\in \mathfrak{G}_{p'}\mathfrak{F}$. (2) $X/O_{p'}(X)$ is a quasisimple group with Sylow $p$-subgroups of order $p$. In particular, if $X=F_p^*(E)$, then $X/O_{p'}(X)$ is a simple group. Theorem B. Let $\mathfrak{F}$ be a solubly saturated formation containing $\mathfrak{U}$ and $E$ a normal subgroup of $G$ with $G/E\in \mathfrak{F}$. Suppose that for any Sylow $p$-subgroup $P$ of $F_p^*(E)$, every cyclic subgroup of $P$ o

We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro-$\mathcal{C}$ completions of the group, where $\mathcal{C}$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $\mathcal{C}$-congruence subgroup property ($\mathcal{C}$-CSP) if its pro-$\mathcal{C}$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $\mathcal{C}$-CSP. In the case where $\mathcal{C}$ is also closed under extensions (for instance the class of all finite $p$-groups for some prime $p$), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the $p$-CSP.

Let $G$ be a finite group. Recall that an $A$-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable $A$-group. Assuming that the commuting graph is connected, we show when the derived length of $G$ is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved.

Let $w=w(x_1,...,x_n)$ be a word, i.e. an element of the free group $F = \langle x_1,...,x_n \rangle$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{ w(x_1,...,x_n) : x_1,...,x_n \in G \}$ of all $w$-values in $G$. Following J. González-Sánchez and B. Klopsch, a group $G$ is $w$-maximal if $|H:w(H)| < |G:w(G)|$ for every $H<G$. In this paper we give new results on $w$-maximal groups, and study the weaker condition in which the previous inequality is not strict. Some applications are given: for example, if a finite group has a solvable (resp. nilpotent) section of size $n$, then it has a solvable (resp. nilpotent) subgroup of size at least $n$.

There are many Lie groups used in physics, including the Lorentz group of special relativity, the spin groups (relativistic and non-relativistic) and the gauge groups of quantum electrodynamics and the weak and strong nuclear forces. Various grand unified theories use larger Lie groups in different attempts to unify some of these groups into something more fundamental. There are also a number of finite symmetry groups that are related to the finite number of distinct elementary particle types. I offer a group-theorist's perspective on these groups, and suggest some ways in which a deeper use of group theory might in principle be useful. These suggestions include a number of options that seem not to be under active investigation at present. I leave open the question of whether they can be implemented in physical theories.

We study Coxeter groups from which there is a natural map onto a symmetric group. Such groups have natural quotient groups related to presentations of the symmetric group on an arbitrary set $T$ of transpositions. These quotients, denoted here by C_Y(T), are a special type of the generalized Coxeter groups defined in \cite{CST}, and also arise in the computation of certain invariants of surfaces. We use a surprising action of $S_n$ on the kernel of the surjection $C_Y(T) \ra S_n$ to show that this kernel embeds in the direct product of $n$ copies of the free group $π_1(T)$ (with the exception of $T$ being the full set of transpositions in $S_4$). As a result, we show that the groups $C_Y(T)$ are either virtually Abelian or contain a non-Abelian free subgroup.

We study the average case complexity of the uniform membership problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the relative primitivity problem, using Shpilrain's recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group.

Let G be a finite solvable permutation group. Then modulo a possibly trivial normal elementary abelian 3-subgroup, some set-stabilizer in G is a 2-group.

We develop a notion of groups that act acylindrically and non-elementarily on simplicial trees, which we call acylindrically arboreal groups. We then prove a complete classification of when graph products of groups and the fundamental groups of certain hyperbolic $3$-manifolds are acylindrically arboreal, and use these classifications to provide examples of acylindrically hyperbolic groups that have actions on trees but have no non-elementary acylindrical actions on trees.

Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups-so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then prove that torsion-free pseudo-fibered groups satisfy a Tits alternative. We conclude by proving that a purely hyperbolic pseudo-fibered group acts on the 2-sphere as a convergence group. This leads to an interesting question if there are examples of pseudo-fibered groups other than 3-manifold groups.

In the paper "Aquino, C., Jiménez, R., Mijangos, M., Morales Meléndez, Q.: On Invariant (co)homology of a group, preprint" are introduced two groups generated by the orbits of an action of a group on another group by automorphisms. One is of group-theoretic nature and the other comes from homology of invariant group chains. In this note are given some properties of the first groups and is studied a natural homomorphism between these groups. More precisely, it is shown that this homomorphism is not injective nor surjective. A description of the kernel is given. Note: there was a previous version with an error in the construction. The error has been corrected now.

By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups.

We will determine all infinite $2$-locally finite groups as well as infinite $2$-groups with planar subgroup graph and show that infinite groups satisfying the chain conditions containing an involution do not have planar embeddings. Also, all connected outer-planar groups and outer-planar groups satisfying the chain conditions are presented. As a result, all planar groups which are direct product of connected groups are obtained.

The Andrews-Curtis conjecture asserts that, for a free group $F_n$ of rank $n$ and a free basis $(x_1,...,x_n)$, any normally generating tuple $(y_1,...,y_n)$ is Andrews-Curtis equivalent to $(x_1,...,x_n)$. This equivalence corresponds to the actions of $\operatorname{Aut}F_n$ and of $F_n$ on normally generating $n$-tuples. The equivalence corresponding to the action of $\operatorname{Aut}F_n$ on generating $n$-tuples is called Nielsen equivalence. The conjecture for arbitrary finitely generated group has its own importance to analyse potential counter-examples to the original conjecture. We study the Andrews-Curtis and Nielsen equivalence in the class of finitely generated groups for which every maximal subgroup is normal, including nilpotent groups and Grigorchuk groups.

We explain and generalise a construction due to Gromov to realise geometric small cancellation groups over graphs of groups as fundamental groups of non-positively curved 2-dimensional complexes of groups. We then give conditions so that the hyperbolicity and some finiteness properties of the small cancellation quotient can be deduced from analogous properties for the local groups of the initial graph of groups.

It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, every countable group $A$ is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if $A$ is finite. In particular, this applies to dessins d'enfants, regarded as finite oriented hypermaps. The proof, involving maximal subgroups of various triangle groups, yields a simple construction of a regular map whose automorphism group contains an isomorphic copy of every finite group.