Podoski and Szegedy [‘On finite groups whose derived subgroup has bounded rank’, Israel J. Math.178 (2010), 51–60] proved that for a finite group $G$ with rank $r$, the inequality $[G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$ holds. In this paper we omit the finiteness condition on $G$ and show that groups with finite derived subgroup satisfy the same inequality. We also construct an $n$-capable group which is not $(n+1)$-capable for every $n\in \mathbf{N}$.

Here we determine up to conjugacy all the maximal subgroups of the finite exceptional group of Lie-type $E_{7}(2)$.

Supplementary materials are available with this article.

We prove that the groups presented by finite convergent monadic rewriting systems with generators of finite order are exactly the free products of finitely many finite groups, thereby confirming Gilman’s conjecture in a special case. We also prove that the finite cyclic groups of order at least three are the only finite groups admitting a presentation by more than one finite convergent monadic rewriting system (up to relabelling), and these admit presentation by exactly two such rewriting systems.

Let $S$ be a finitely generated pro-$p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$. In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$. For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

We construct cocompact lattices Γ'0 < Γ0 in the group G = PGLd$({\mathbb{F}_q(\!(t)\!)\!})$ which are type-preserving and act transitively on the set of vertices of each type in the building Δ associated to G. These lattices are commensurable with the lattices of Cartwright–Steger Isr. J. Math.103 (1998), 125–140. The stabiliser of each vertex in Γ'0 is a Singer cycle and the stabiliser of each vertex in Γ0 is isomorphic to the normaliser of a Singer cycle in PGLd(q). We show that the intersections of Γ'0 and Γ0 with PSLd$({\mathbb{F}_q(\!(t)\!)\!})$ are lattices in PSLd$({\mathbb{F}_q(\!(t)\!)\!})$, and identify the pairs (d, q) such that the entire lattice Γ'0 or Γ0 is contained in PSLd$({\mathbb{F}_q(\!(t)\!)\!})$. Finally we discuss minimality of covolumes of cocompact lattices in SL3$({\mathbb{F}_q(\!(t)\!)\!})$. Our proofs combine the construction of Cartwright–Steger Isr. J. Math.103 (1998), 125–140 with results about Singer cycles and their normalisers, and geometric arguments.

Given two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces $X,Y,Z$ and derive a condition, called the (${\it\delta}$-polynomial) path lifting property, such that coarse embeddability of $X,Y$ and $Z$ implies coarse embeddability of $X\wr _{Z}Y$. We also give bounds on the compression of $X\wr _{Z}Y$ in terms of ${\it\delta}$ and the compressions of $X,Y$ and $Z$.

We consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to Gromov’s well-known constructions, and include for example a ‘density model’ for groups of intermediate rank. The main novelty is the higher rank nature of the random groups. They are randomizations of certain families of lattices in algebraic groups (of rank 2) over local fields.

Let $G$ be a finite group, let ${\it\pi}(G)$ be the set of prime divisors of $|G|$ and let ${\rm\Gamma}(G)$ be the prime graph of $G$. This graph has vertex set ${\it\pi}(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that ${\rm\Gamma}(G)={\rm\Gamma}(H)$.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ is a properly embedded $\pi _1$-injective surface in a compact 3-manifold $M$, then $\pi _1S$ is separable in $\pi _1M$.

Let $W$ be an extended affine Weyl group. We prove that the minimal length elements $w_{{\mathcal{O}}}$ of any conjugacy class ${\mathcal{O}}$ of $W$ satisfy some nice properties, generalizing results of Geck and Pfeiffer [On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), 79–94] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some $p$-adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra $H$. We prove that $T_{w_{{\mathcal{O}}}}$, where ${\mathcal{O}}$ ranges over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H,H]$. We also introduce the class polynomials, which play a crucial role in the study of affine Deligne–Lusztig varieties He [Geometric and cohomological properties of affine Deligne–Lusztig varieties, Ann. of Math. (2) 179 (2014), 367–404].

Assume that $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}m$ and $n$ are two positive integers which do not divide each other. If the set of conjugacy class sizes of primary and biprimary elements of a group $G$ is $\{1, m, n, mn\}$, we show that up to central factors $G$ is a $\{p,q\}$-group for two distinct primes $p$ and $q$.

We introduce a family of infinite nonamenable discrete groups as an interpolation of the Higman–Thompson groups by using the topological full groups of the groupoids defined by $\beta $-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras, and realized as groups of piecewise-linear functions on the unit interval in the real line if the $\beta $-expansion of $1$ is finite or ultimately periodic. We also classify them by a number-theoretical property of $\beta $.

We prove the existence of certain rationally rigid triples in ${E}_{8}(p)$ for good primes $p$ (i.e. $p>5$) thereby showing that these groups occur as Galois groups over the field of rational numbers. We show that these triples arise from rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic zero. As a byproduct of the proof, we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a short list of possible overgroups of regular unipotent elements in simple exceptional groups.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a finite 2-group. If $G$ is of coclass 2 or $(G,Z(G))$ is a Camina pair, then $G$ admits a noninner automorphism of order 2 or 4 leaving the Frattini subgroup elementwise fixed.

The present paper is related to some recent studies in Abdollahi and Russo [‘On a problem of P. Hall for Engel words’, Arch. Math. (Basel) 97 (2011), 407–412] and Fernández-Alcober et al. [‘A note on conciseness of Engel words’, Comm. Algebra 40 (2012), 2570–2576] on the position of the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$-Engel marginal subgroup $E^*_n(G)$ of a group $G$, when $n=3,4$. Describing the size of $E^*_n(G)$ for $n=3,4$, we show some generalisations of classical results on the partial margins of $E^*_3(G)$ and $E^*_4(G)$.

In this paper we prove that every group with at most 26 normalisers is soluble. This gives a positive answer to Conjecture 3.6 in the author’s paper [On groups with a finite number of normalisers’, Bull. Aust. Math. Soc.86 (2012), 416–423].

Given a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$, we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$. Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory.

Let $G(q)$ be a finite Chevalley group, where $q$ is a power of a good prime $p$, and let $U(q)$ be a Sylow $p$-subgroup of $G(q)$. Then a generalized version of a conjecture of Higman asserts that the number $k(U(q))$ of conjugacy classes in $U(q)$ is given by a polynomial in $q$ with integer coefficients. In [S. M. Goodwin and G. Röhrle, J. Algebra 321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of $k(U(q))$. By implementing it into a computer program using $\mathsf{GAP}$, they were able to calculate $k(U(q))$ for $G$ of rank at most five, thereby proving that for these cases $k(U(q))$ is given by a polynomial in $q$. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of $k(U(q))$ for finite Chevalley groups of rank six and seven, except $E_7$. We observe that $k(U(q))$ is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write $k(U(q))$ as a polynomial in $q-1$, then the coefficients are non-negative.

Under the assumption that $k(U(q))$ is a polynomial in $q-1$, we also give an explicit formula for the coefficients of $k(U(q))$ of degrees zero, one and two.

In this paper, we establish that complete Kac–Moody groups over finite fields are abstractly simple. The proof makes essential use of Mathieu and Rousseau’s construction of complete Kac–Moody groups over fields. This construction has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.