We compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

Let G be a finite group with normal subgroup N. A subgroup K of G is a partial complement of N in G if N and K intersect trivially. We study the partial complements of N in the following case: G is soluble, N is a product of minimal normal subgroups of G, N has a complement in G, and all such complements are G-conjugate.

We show that if G is a group and A⊂G is a finite set with ∣A2∣≤K∣A∣, then there is a symmetric neighbourhood of the identity S such that Sk⊂A2A−2 and ∣S∣≥exp (−KO(k))∣A∣.

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn,1.

In this paper we construct the maximal subgroups of geometric type of the orthogonal groups in dimension d over GF(q) in O(d3+d2log q+log qlog log q) finite field operations.

We give a new elementary construction of Ree's family of finite simple groups of type 2G2, which avoids the need for the machinery of Lie algebras and algebraic groups. We prove that the groups we construct are simple of order q3(q3 + 1)(q − 1) and act doubly transitively on an explicit set of q3 + 1 points, where q = 32k+1. Moreover, our construction is practical in the sense that generators for the groups and many of their maximal subgroups may easily be obtained.

For each finite real reflection group W, we identify a copy of the type-W simplicial generalized associahedron inside the corresponding simplicial permutahedron. This defines a bijection between the facets of the generalized associahedron and the elements of the type-W non-crossing partition lattice that is more tractable than previous such bijections. We show that the simplicial fan determined by this associahedron coincides with the Cambrian fan for W.

We observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc.12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc.12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc.74(3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p′-elements, a formation Pp,p′, for each prime p. We show that P2,2′ (which contains the odd-order groups) is properly contained in the solvable formation.

We consider finite p-groups G in which every cyclic subgroup has at most p conjugates. We show that the derived subgroup of such a group has order at most p2. Further, if the stronger condition holds that all subgroups have at most p conjugates then the central factor group has order p4 at most.

We compute the conjugacy classes of elements and the character tables of the maximal parabolic subgroups of the simple Ree groups 2F4(q2). For one of the maximal parabolic subgroups, we find an irreducible character of the unipotent radical that does not extend to its inertia subgroup.

Let G be a reductive p-adic group. Given a compact-mod-center maximal torus S⊂G and sufficiently regular character χ of S, one can define, following Adler, Yu and others, a supercuspidal representation π(S,χ) of G. For S unramified, we determine when π(S,χ) is generic, and which generic characters it contains.

Let ℓ be a prime number. It is not known whether every finite ℓ-group of rank n≥1 can be realized as a Galois group over with no more than n ramified primes. We prove that this can be done for the (minimal) family of finite ℓ-groups which contains all the cyclic groups of ℓ-power order and is closed under direct products, (regular) wreath products and rank-preserving homomorphic images. This family contains the Sylow ℓ-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not ℓ. On the other hand, it does not contain all finite ℓ-groups.

In this paper, four new discreteness criteria for isometric groups on complex hyperbolic spaces are proved, one of which shows that the Condition C hypothesis in Cao [‘Discrete and dense subgroups acting on complex hyperbolic space’, Bull. Aust. Math. Soc.78 (2008), 211–224, Theorem 1.4] is removable; another shows that the parabolic condition hypothesis in Li and Wang [‘Discreteness criteria for Möbius groups acting on II’, Bull. Aust. Math. Soc.80 (2009), 275–290, Theorem 3.1] is not necessary.

We use the technique of Fischer matrices to write a program to produce the character table of a group of shape (2×2.G):2 from the character tables of G, G:2, 2.G and 2.G:2.

Let R be a ring, S a strictly ordered monoid, and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

We decompose the restriction of ramified principal series representations of the $p$ -adic group $\text{GL}\left( 3,\,\text{k} \right)$ to its maximal compact subgroup $K\,=\,\text{GL}\left( 3,\,\mathcal{R} \right)$ . Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$ . We establish several irreducibility results and illustrate the decomposition with some examples.

Marques-Smith and Sullivan [‘Partial orders on transformation semigroups’, Monatsh. Math.140 (2003), 103–118] studied various properties of two partial orders on P(X), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X. One partial order was the ‘containment order’: namely, if α,β∈P(X) then α⊆β means xα=xβ for all x∈dom α, the domain of α. The other order was the so-called ‘natural order’ defined by Mitsch [‘A natural partial order for semigroups’, Proc. Amer. Math. Soc.97(3) (1986), 384–388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I(X) and the partial Baer–Levi semigroup PS(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements.

An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg–MacLane space.

Let G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.

A group G is called morphic if every endomorphism α:G→G for which Gα◃G satisfies G/Gα≅ker (α). Call an endomorphism α∈end(G) regular if αβα=α for some β∈end(G), and call α unit regular if β can be chosen to be an automorphism of G. The main purpose of this paper is to prove the following group-theoretic analogue of a theorem of Ehrlich: if G is a morphic group, an endomorphism α:G→G for which Gα◃G is unit regular if and only if it is regular. As an application, a cancellation theorem is proved that characterizes the morphic groups among those with regular endomorphism monoids.