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No group has exactly one or two nonpower subgroups. We classify groups containing exactly three nonpower subgroups and show that there is a unique finite group with exactly four nonpower subgroups. Finally, we show that given any integer k greater than
$4$
, there are infinitely many groups with exactly k nonpower subgroups.
We show that if G is an amenable group and H is a hyperbolic group, then the free product
$G\ast H$
is weakly amenable. A key ingredient in the proof is the fact that
$G\ast H$
is orbit equivalent to
$\mathbb{Z}\ast H$
.
Given groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered.
Let $m\leqslant n\in \mathbb {N}$, and $G\leqslant \operatorname {Sym}(m)$ and $H\leqslant \operatorname {Sym}(n)$. In this article, we find conditions enabling embeddings between the symmetric R. Thompson groups ${V_m(G)}$ and ${V_n(H)}$. When $n\equiv 1 \mod (m-1)$, and under some other technical conditions, we find an embedding of ${V_n(H)}$ into ${V_m(G)}$ via topological conjugation. With the same modular condition, we also generalize a purely algebraic construction of Birget from 2019 to find a group $H\leqslant \operatorname {Sym}(n)$ and an embedding of ${V_m(G)}$ into ${V_n(H)}$.
Let $K$ be a subgroup of a finite group $G$. The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. Assume that $Pr(K,G)\geq \epsilon$ for some fixed $\epsilon >0$. We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indices $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$-bounded. This extends the well-known theorem, due to P. M. Neumann, that covers the case where $K=G$. We deduce a number of corollaries of this result. A typical application is that if $K$ is the generalized Fitting subgroup $F^{*}(G)$ then $G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $\epsilon$-bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$, or a Sylow subgroup, etc.
A Grigorchuk–Gupta–Sidki (GGS)-group is a subgroup of the automorphism group of the p-regular rooted tree for an odd prime p, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here, we extend the result to nontorsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.
We generalize results of Thomas, Allcock, Thom–Petersen, and Kar–Niblo to the first $\ell ^{2}$-Betti number of quotients of certain groups acting on trees by subgroups with free actions on the edge sets of the graphs.
To each automorphism of a spherical building, there is a naturally associated opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber, then the automorphism is called domestic. In this paper, we give the complete classification of domestic automorphisms of split spherical buildings of types
$\mathsf {E}_6$
,
$\mathsf {F}_4$
, and
$\mathsf {G}_2$
. Moreover, for all split spherical buildings of exceptional type, we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most two distinguished orbits encircled. Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.
Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a continuous representation
$\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$
such that every open subgroup H of G, together with the restriction
$\theta \vert _H$
, satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that if a 1-smooth pro-p group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields. Finally, we ask whether 1-smooth pro-p groups satisfy a “Tits’ alternative.”
For any (Hausdorff) compact group G, denote by
$\mathrm{cp}(G)$
the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that
$\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$
, where F is the FC-centre of G and H is isoclinic to F with
$\mathrm{cp}(F)=\mathrm{cp}(H)$
whenever
$\mathrm{cp}(G)>0$
. In addition, we prove that a compact group G with
$\mathrm{cp}(G)>\tfrac {3}{40}$
is either solvable or isomorphic to
$A_5 \times Z(G)$
, where
$A_5$
denotes the alternating group of degree five and the centre
$Z(G)$
of G contains the identity component of G.
A subgroup H of a group G is said to be contranormal in G if the normal closure of H in G is equal to G. In this paper, we consider groups whose nonmodular subgroups (of infinite rank) are contranormal.
Given a finite group G with a normal subgroup N, the simple graph
$\Gamma _{\textit {G}}( \textit {N} )$
is a graph whose vertices are of the form
$|x^G|$
, where
$x\in {N\setminus {Z(G)}}$
and
$x^G$
is the G-conjugacy class of N containing the element x. Two vertices
$|x^G|$
and
$|y^G|$
are adjacent if they are not coprime. We prove that, if
$\Gamma _G(N)$
is a connected incomplete regular graph, then
$N= P \times {A}$
where P is a p-group, for some prime p,
$A\leq {Z(G)}$
and
$\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$
.
A subgroup H of a group G is pronormal in G if each of its conjugates
$H^g$
in G is conjugate to it in the subgroup
$\langle H,H^g\rangle $
; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.
Morgan and Parker proved that if G is a group with
${\textbf{Z}(G)} = 1$
, then the connected components of the commuting graph of G have diameter at most
$10$
. Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a
$2$
-Frobenius group, and if the commuting graph of G is connected, then its diameter is at most
$8$
. We prove that the hypothesis
$Z (G) = 1$
in these results can be replaced with
$G' \cap {\textbf{Z}(G)} = 1$
. We also prove that if G is solvable and
$G/{\textbf{Z}(G)}$
is either a Frobenius group or a
$2$
-Frobenius group, then the commuting graph of G is disconnected.
We extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.
The paper gives a simple proof of Graev’s theorem (asserting that the free product of Hausdorff topological groups is Hausdorff) for a particular case which includes the countable case of
$k_\omega $
-groups and the countable case of Lindelöf P-groups. For this it is shown that in these particular cases the topology of the free product of Hausdorff topological groups coincides with the
$X_0$
-topology in the Mal’cev sense, where X is the disjoint union of the topological groups identifying their units.
We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.
Let $G(n)={\textrm {Sp}}(n,1)$ or
${\textrm {SU}}(n,1)$. We classify conjugation orbits of generic pairs of loxodromic elements in
$G(n)$. Such pairs, called ‘nonsingular’, were introduced by Gongopadhyay and Parsad for
${\textrm {SU}}(3,1)$. We extend this notion and classify
$G(n)$-conjugation orbits of such elements in arbitrary dimension. For
$n=3$, they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed (genus
$g \geq 2$) oriented surface into
$G(3)$.