Let G be a finite non-solvable group. We prove that there exists a proper subgroup A of G such that G is the product of three conjugates of A, thus replacing an earlier upper bound of 36 with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group G with a BN -pair and a finite Weyl group W satisfies G = (Bn 0 B) 2 = BB n 0 B where n 0 is any preimage of the longest element of W . The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of B in G. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.

Abstract. We prove that every finite simple group G of Lie type satisfies G = U U − U U − where U is a unipotent Sylow subgroup of G and U − is its opposite. We also characterize the cases for which G = U U − U . These results are best possible in terms of the number of conjugates of U in the above factorizations.

Héthelyi and Külshammer showed that the number of conjugacy classes k(G) of any solvable finite group G whose order is divisible by the square of a prime p is at least (49p + 1)/60. Here an asymptotic generalization of this result is established. It is proved that there exists a constant c > 0 such that for any finite group G whose order is divisible by the square of a prime p we have k(G) ≥ cp.

Let G be a finite reductive group defined over Fq, with q a power of a prime p. Motivated by a problem recently posed by C. Curtis, we first develop an algorithm to express each element of G into a canonical form in terms of a refinement of a Bruhat decomposition, and we then use the output of the algorithm to explicitly determine the structure constants of the endomorphism algebra of a Gelfand-Graev representation of G when G = PGL3(q) for an arbitrary prime p, and when G = SO5(q) for p odd.Let G be the fixed point subgroupḠ F of a reductive algebraic groupḠ under a Frobenius endomorphism F . A special role in the representation theory of finite reductive groups is played by Gelfand-Graev representations. These are certain representations induced from a linear character of a maximal unipotent subgroup of G. If the center of the ambient algebraic groupḠ is connected, then a Gelfand-Graev representation Γ is unique up to conjugacy and the irreducible components of Γ were described in [DL76]. In general, these representations were parametrized and decomposed in [DM91, Section 14] and [DLM92, Section 3].Denote by H the Hecke algebra (that is, the G-endomorphism algebra) of the module affording Γ. Since Γ is multiplicity free [Car85, Theorem 8.1.3], the algebra H is abelian. C. Curtis parametrized in [Cur93] the irreducible representations of H by pairs (T, θ) where T =T F for an F -stable maximal torusT ofḠ and where θ is an irreducible character of T . Each irreducible representation f T,θ of the algebra H is shown to have the following factorization [Cur93, Theorem 4.2], f T,θ =θ • f T , where f T : H →Q ℓ T is a homomorphism of algebras andθ is the linear extension of θ to the group algebraQ ℓ T . The homomorphism f T is independent of θ. Curtis' homomorphism f T was considered in [BK08] in the context of ℓ-modular Gelfand-Graev representations with ℓ = p.Here it is shown that if ℓ does not divide the order of the Weyl group of G, then the behavior of such endomorphism algebras is generic along all prime powers q.In order to obtain a constructive description of each f T , one needs to consider structure constants of H with respect to some basis. Building on work of Kawanaka [Kaw77] and Deodhar [Deo85], Curtis described in [Cur15] an algorithm for obtaining structure constants for H with respect to a standard basis parametrized by certain elements in N G (T ). In general, the determination of the structure

For a connected reductive algebraic group G defined over an algebraically closed field of characteristic p the sheets of conjugacy classes have been parametrized by G. Carnovale and F. Esposito when p is good for G. We show that the method is independent of characteristic and that a similar parametrization is possible for all p. MSC 2010. 14L10, 14L35, 14L40.