Let R be an integral domain and A a symmetric cellular algebra over R with a cellular basis {CλS,T∣λ∈Λ,S,T∈M(λ)}. We construct an ideal L(A) of the centre of A and prove that L(A) contains the so-called Higman ideal. When R is a field, we prove that the dimension of L(A)is not less than the number of nonisomorphic simple A-modules.

Suppose that p is 3,5,7,11 or 13. We classify the radical p-chains of the Monster 𝕄 and verify the Alperin weight conjecture and Uno’s reductive conjecture for 𝕄, the latter being a refinement of Dade’s reductive conjecture and the Isaacs–Navarro conjecture.

Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported,K-biequivariant functions f:G(F)→End   V. These Hecke algebras were first considered by Barthel and Livné for GL 2. They play a role in the recent mod p andp-adic Langlands correspondences for GL 2 (ℚp) , in generalisations of Serre’s conjecture on the modularity of mod p Galois representations, and in the classification of irreducible mod p representations of unramified p-adic reductive groups.

Let G be a finite group. If G has a cyclic Sylow 2-subgroup, then G has the same number of irreducible rational-valued characters as of rational conjugacy classes. These numbers need not be the same even if G has Klein Sylow 2-subgroups and a normal 2-complement.

In this paper we refine and extend the applicability of the algorithms in Bates and Rowley (Arch. Math. 92 (2009) 7–13) for computing part of the normalizer of a 2-subgroup in a black-box group.

Supplementary materials are available with this article.

We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras, including a generalization of the concept of aW-graph to the situation of complex reflection groups. We then use these techniques to find models for all irreducible representations in the case of complex reflection groups of dimension at most three. Using these models we are able to verify some important conjectures on the structure of Hecke algebras.

A finite group is called a CH-group if for every x,y∈G∖Z(G), xy=yx implies that . Applying results of Schmidt [‘Zentralisatorverbände endlicher Gruppen’, Rend. Sem. Mat. Univ. Padova44 (1970), 97–131] and Rebmann [‘F-Gruppen’, Arch. Math. 22 (1971), 225–230] concerning CA-groups and F-groups, the structure of CH-groups is determined, up to that of CH-groups of prime-power order. Upper bounds are found for the derived length of nilpotent and solvable CH-groups.

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.

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.

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 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.

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.

We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n ≥ p2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if n ≥ p2 + p and has rank 2 if p2 ≤ n < p2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano.

Finite groups with abelian Sylow p-subgroups for certain primes p are characterized in terms of arithmetical properties of commutators.

Colmez has given a recipe to associate a smooth modular representation Ω(W) of the Borel subgroup of GL2(Qp) to a -representation W of by using Fontaine’s theory of (φ,Γ)-modules. We compute Ω(W) explicitly and we prove that if W is irreducible and dim (W)=2, then Ω(W) is the restriction to the Borel subgroup of GL2(Qp) of the supersingular representation associated to W by Breuil’s correspondence.

Let G be a finite group and let δ(G) be the number of prime order subgroups of G. We determine the groups G with the property δ(G)≥∣G∣/2−1, extending earlier work of C. T. C. Wall, and we use our classification to obtain new results on the generation of near-rings by units of prime order.