A sufficient condition for the existence of HNN-extensions in the class of groups of odd exponent $n \gg 1$ is given in the following form. Let $Q$ be a group of odd exponent $n > 2^{48}$ and $\mathcal G$ be an HNN-extension of $Q$. If $A \in \mathcal G$ then let $\mathcal F(A)$ denote the maximal subgroup of $Q$ which is normalized by $A$. By $τ_A$ denote the automorphism of $\mathcal F(A)$ which is induced by conjugation by $A$. Suppose that for every $A \in \mathcal G$, which is not conjugate to an element of $Q$, the group $<τ_A, \mathcal F(A)>$ has exponent $n$ and, in addition, equalities $A^{-k} q_0 A^{k} = q_k$, where $q_k \in Q$ and $k =0, 1, ..., [2^{-16}n]$ ($[2^{-16}n]$ is the integer part of $2^{-16}n$), imply that $q_0 \in \mathcal F(A)$. Then the group $Q$ naturally embeds in the quotient $\mathcal G / \mathcal G^n$, that is, there exists an analog of the HNN-extension $\mathcal G$ of $Q$ in the class of groups of exponent $n$.
We prove the following version of Milnor's theorem on solvable groups of exponential growth: A finitely generated solvable group which is not polycyclic contains an ascending HNN extension. Consequently, a finitely generated solvable group is ERF iff it is polycyclic.
If a finitely generated torsion free group K has the property that all finitely generated subgroups S of K are either small or have growth constant bounded uniformly away from 1 then a non proper HNN extension G of K, that is a semidirect product of K by the integers, has the same property. Here small means cyclic or, if the automorphism has no periodic conjugacy classes, free abelian of bounded rank.
The main result in this paper is the failure of the finitely generated intersection property (FGIP) of ascending HNN extensions of non-cyclic finite rank free groups. This class of group consists of free-by-cyclic groups and properly ascending HNN extensions of free groups. We also give a sufficient condition for the failure of the FGIP in the context of relative hyperbolicity, we apply this to free-by-cyclic groups of exponential growth.
We study ascending HNN-extensions $G$ of finitely generated free abelian groups: examples of such $G$ include soluble Baumslag-Solitar groups and fundamental groups of orientable prime $3$-manifolds modelled on Sol geometry. In particular, we study the elliptic subgroup $A \leq G$, consisting of all elements that stabilise a point in the Bass-Serre tree of $G$. We consider the density of $A$ with respect to ball counting measures corresponding to finite generating sets of $G$, and we show that $A$ is exponentially negligible in $G$ with respect to such sequences of measures. As a consequence, …
We prove that for any prime $p\geq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $\mathcal{L}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true and the growth rate of $BS(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group $\mathcal{L}_2$ is equal to the golden ratio $(1+\sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidire…
We prove that every noncyclic subgroup of a free $m$-generator Burnside group $B(m,n)$ of odd exponent $n \gg 1$ contains a subgroup $H$ isomorphic to a free Burnside group $B(\infty,n)$ of exponent $n$ and countably infinite rank such that for every normal subgroup $K$ of $H$ the normal closure $<K >^{B(m,n)}$ of $K$ in $B(m,n)$ meets $H$ in $K$. This implies that every noncyclic subgroup of $B(m,n)$ is SQ-universal in the class of groups of exponent $n$.
Let $G$ be a finite group of odd order admitting an involutory automorphism $ϕ$. We obtain two results bounding the exponent of $[G,ϕ]$. Denote by $G_{-ϕ}$ the set $\{[g,ϕ]\,\vert\, g\in G\}$ and by $G_ϕ$ the centralizer of $ϕ$, that is, the subgroup of fixed points of $ϕ$. The obtained results are as follows.1. Assume that the subgroup $\langle x,y\rangle$ has derived length at most $d$ and $x^e=1$ for every $x,y\in G_{-ϕ}$. Suppose that $G_ϕ$ is nilpotent of class $c$. Then the exponent of $[G,ϕ]$ is $(c,d,e)$-bounded.2. Assume that $G_ϕ$ has rank $r$ and $x^e=1$ for each $x\in G_{-ϕ}$. Then…
An odd Coxeter group $W$ is one which admits a Coxeter system $(W,S)$ for which all the exponents $m_{ij}$ are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs $\mathcal{V}_{(W,S)}$ are trees. It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same multiset of finite exponents. In particular, each group in this family is isomorphic to a group that admits a Coxeter system whose associated labeled graph is a star shaped tree. We give the complete de…
We give a criterion for a finitely generated odd-angled Coxeter group to have a proper finite index subgroup generated by reflections. The answer is given in terms of the least prime divisors of the exponents of the Coxeter relations.
We develop yet another technique to present the free Burnside group $B(m,n)$ of odd exponent $n$ with $m\ge2$ generators as a group satisfying a certain iterated small cancellation condition. Using the approach, we provide a reasonably accessible proof that $B(m,n)$ is infinite with a moderate bound $n > 2000$ on the odd exponent $n$.
Let $G$ be an odd order nilpotent group with class 2 and $e$ denotes the exponent of its commutator subgroup. Let $e=p_1^{r_1}p_2^{r_2}... p_s^{r_s}$, where $p_i$'s are odd primes and $r_i$'s are non-negative integers. Then there are at least $r_1+r_2+... +r_s$ non-isomorphic nilpotent groups with class two and the order of each of the group is equal to the order of $G$.
We study the strong, weak, and special amalgamation bases in the varieties of nilpotent groups of class two and exponent n, where n is odd. The main result is a characterization of the special amalgamation bases for these varieties. We also characterize the weak and strong bases. For special amalgamation bases, we show that there are groups which are special bases in varieties of finite exponent but not in the variety of all nil-2 groups, whereas for weak and strong bases we show this is not the case. We also show that in these varieties, as well as the variety of all nil-2 groups, a group has…
We consider the capability of $p$ groups of class two and odd prime exponent. We use linear algebra and counting arguments to establish a number of new results. In particular, we settle the 4-generator case, and prove a sufficient condition based on the ranks of $G/Z(G)$ and $[G,G]$.
Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.
A longstanding problem attributed to I. Schur says that for a finite group $G$, the exponent of the second homology group $H_2(G, \mathbb{Z})$ divides the exponent of $G$. In this paper, we prove this conjecture for finite nilpotent groups of odd exponent and of nilpotency class 5, $p$-central metabelian $p$ groups, and groups considered by L. E . Wilson in \cite{LEW}. Moreover, we improve several bounds given by various authors.
We extend the notion of free $p$-central groups for odd primes $p$ to the case $p=2$ by introducing a variant of the lower $p$-central series. This enables us to calculate Schur multipliers of free $p$-central groups. We also prove that for any $p$-central group the exponent of its Schur multiplier divides the exponent of the group, and determine its exponential rank.
We study the commuting graph on elements of odd prime order in finite simple groups. The results are used in a forthcoming paper describing the structure of Bruck loops and Bol loops of exponent 2.
We prove that an isomorphism between saturated fusion systems over the same finite p-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if p is odd, and on the abelian subgroups of the hyperfocal subgroup of exponent at most 4 if p = 2. For odd p, this has implications for mod p group cohomology.
A group is called capable if it is a central factor group. We consider the capability of finite groups of class two and exponent $p$, $p$ an odd prime. We restate the problem of capability as a problem about linear transformations, which may be checked explicitly for any specific instance of the problem. We use this restatement to derive some known results, and prove new ones. Among them, we reduce the general problem to an oft-considered special case, and prove that a 3-generated group of class 2 and exponent $p$ is either cyclic or capable.
It is a longstanding conjecture that for a finite group $G$, the exponent of the second homology group $H_2(G, \mathbb{Z})$ divides the exponent of $G$. In this paper, we prove this conjecture for $p$-groups of class at most $p$, finite nilpotent groups of odd exponent and of nilpotency class 5, $p$-central metabelian $p$-groups, and groups considered by L. E . Wilson in \cite{LEW}. Moreover, we improve several bounds given by various authors. We achieve most of our results using an induction argument.
It is well known that the positive degree cohomology of a finite group G is annihilated by |G|. We improve on this bound in the case of odd degree elements in the integer cohomology ring and show that $e_{odd}(G)$, the exponent of the $\oplus_{k=0}^{\infty} H^{2k+1}(G,\mathbb{Z})$ satisfies $e_{odd}(G)^2$ divides 2|G| and in particular $e_{odd}(G) \leq \sqrt{2|G|}.$ We also provide examples to show this bound for $e_{odd}(G)$ is sharp as a general bound over all finite groups G. The result comes from a fact about zero divisors having "complementary exponent" which we prove using duality in Tat…
We consider the capability of $p$-groups of class two and odd prime exponent. The question of capability is shown to be equivalent to a statement about vector spaces and linear transformations, and using the equivalence we give proofs of some old results and several new ones. In particular, we establish a number of new necessary and new sufficient conditions for capability, including a sufficient condition based only on the ranks of $G/Z(G)$ and $[G,G]$. Finally, we characterise the capable groups among the 5-generated groups in this class.
We construct an embedding of a free Burnside group $B(m,n)$ of odd $n > 2^{48}$ and rank $m >1$ in a finitely presented group with some special properties. The main application of this embedding is an easy construction of finitely presented non-amenable groups without noncyclic free subgroups (which provides a finitely presented counterexample to the von Neumann problem on amenable groups). As another application, we construct weakly finitely presented groups of odd exponent $n \gg 1$ which are not locally finite.
If $p$ is an odd prime, then we prove that $\e(H_2(G,\mathbb{Z})) \mid p\ \e(G)$ for $p$ groups of class 7. We prove the same for $p$ groups of class at most $p+1$ with $\e(Z(G))=p$. We also prove Schurs conjecture if $\e(G/Z(G))$ is $2,3$ or $6$. Furthermore we prove that if $G$ is a solvable group of derived length $d$ and $\e(G)=p$, then $\e(H_2(G,\mathbb{Z})) \mid (\e(G))^{d-1}$. We also show that if $G$ is a finite $2$ or $3$ generator group of exponent 5, then $\e(H_2(G,\mathbb{Z})) \mid (\e(G))^2$.
Given any group $G$, the multiple holomorph $\mathrm{NHol}(G)$ is the normalizer of the holomorph $\mathrm{Hol}(G) = ρ(G)\rtimes \mathrm{Aut}(G)$ in the group of all permutations of $G$, where $ρ$ denotes the right regular representation. The quotient $T(G) = \mathrm{NHol}(G)/\mathrm{Hol}G)$ has order a power of $2$ in many of the known cases, but there are exceptions. We shall give a new method of constructing elements (of odd order) in $T(G)$ when $G=A\rtimes C_d$, where $A$ is a group of finite exponent coprime to $d$ and $C_d$ is the cyclic group of order $d$.
In this paper, we study some properties of the outer automorphism group of free Burnside groups of large odd exponent. In particular, we prove that it contains free and free abelian subgroups.
We show that free Burnside groups of sufficiently large odd exponent are non--amenable in a certain strong sense, more precisely, their left regular representations are isolated from the trivial representation uniformly on finite generating sets. This result is applied to the solution of a strong version of the von Neumann -- Day problem concerning amenability of groups without non--abelian free subgroups. As another consequence, we obtain that the above--mentioned groups are of uniform exponential growth.
We introduce a special class of powerful $p$-groups that we call powerfully nilpotent groups that are finite $p$-groups that possess a central series of a special kind. To these we can attach the notion of a powerful nilpotence class that leads naturally to a classification in terms of an `ancestry tree' and powerful coclass. We show that there are finitely many powerfully nilpotent $p$-groups of each given powerful coclass and develop some general theory for this class of groups. We also determine the growth of powerfully nilpotent groups of exponent $p^{2}$ and order $p^{n}$ where $p$ is odd…
This paper studies the MOR cryptosystem, using the automorphism group of the extra-special $p$-group of exponent $p$, for an odd prime $p$. Similar results can be obtained for extra-special $p$-groups of exponent $p^2$ and for the even prime.
Let $p$ be a an odd prime and let $G$ be a finite $p$-group with cyclic commutator subgroup $G'$. We prove that the exponent and the abelianization of the centralizer of $G'$ in $G$ are determined by the group algebra of $G$ over any field of characteristic $p$. If, additionally, $G$ is $2$-generated then almost all the numerical invariants determining $G$ up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of $G'$ is determined. These claims are known to be false for $p=2$.
