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In this article, we define the notion of Brauer-Clifford group for H-locally finite (S, H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is an H-locally finite commutative H-module algebra over a commutative noetherian ring k. This is the situation that arises in applications with connections to algebraic geometry. This Brauer-Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category. [ABSTRACT FROM AUTHOR]

For a formal scheme over a complete discrete valuation ring with a good action of a finite group, we define equivariant motivic integration, and we prove a change of variable formula for that. To do so, we construct and examine an induced group action on the Greenberg scheme of such a formal scheme. Using this equivariant motivic integration, we define an equivariant volume Poincaré series, from which we deduce Denef and Loeser's motivic zeta function including the action of the profinite group of roots of unity. [ABSTRACT FROM AUTHOR]

Letbe a connected reductive algebraic group defined over an algebraically closed field. Letbe a maximal torus ofand letbe a Borel subgroup ofcontaining. LetWdenote the Weyl group ofand letẇdenote an element in the preimage of the natural map fromtoW. We investigate what conditions are necessary and sufficient for the double cosets(called Bruhat cells) to contain only regular elements. An application is provided on how this information about Bruhat cells can be used to determine character values for certain Hecke algebras. [ABSTRACT FROM PUBLISHER]

LetGbe a finite group,Ea normal subgroup ofGandpa fixed prime. We say thatEisp-hypercyclically embeddedinGif everyp-G-chief factor ofEis cyclic. A subgroupHofGis said to satisfy Π-propertyinGif |G∕K:NG∕K((H∩L)K∕K)| is aπ((H∩L)K∕K)-number for any chief factorL∕KinG; we say thatHhas Π -property inGifH∩Oπ(H)(G) has Π-property inG. In this paper, we prove thatEisp-hypercyclically embedded inGif and only if some classes ofp-subgroups ofEhave Π -property inG. Some recent results are extended. [ABSTRACT FROM AUTHOR]

We show that if |Aut(G)| is odd, then without loss of generality we may assume the only primes dividing |G| are those dividing |Aut(G)|. We also classify finite abelian groups A such that |Aut(A)| = |A| and finite abelian groups A such that |Aut(A)| p = |A| p , where p is a prime dividing |A|. [ABSTRACT FROM PUBLISHER]

Let G be a finite group and ℱ a formation. A subgroup H is called ℱ-quasinormal in G if there exists a quasinormal subgroup T of G such that HT is quasinormal in G and (H ∩ T)H G /H G is contained in the ℱ-hypercenter of G/H G . In this article, we study the structure of finite groups by using ℱ-quasinormal subgroups and prove that: Let ℱ be a saturated formation containing 𝒰 and G be a group with a normal subgroup H such that G/H ∈ ℱ. If every maximal subgroup of every noncyclic Sylow subgroup of F*(H) having no supersolvable supplement in G is 𝒰-quasinormal in G, then G ∈ ℱ. [ABSTRACT FROM PUBLISHER]

We give several characterizations of Zimmermann's uniform chain condition on matrix subgroups. In particular, we prove that a family of modules satisfies it precisely when the direct sum of their pure injective envelopes splits in their direct product. We also prove that a weaker non-uniform version of this condition is equivalent to Σ-algebraic compactness of reduced products. [ABSTRACT FROM PUBLISHER]

This work examines the possible projectivity of 2-modular block parts of nonprojective Lefschetz characters over 2-local geometries of several sporadic groups. Previously known results on M12, J2, and HS are mentioned for completeness. The main new results are on the sporadic groups Suz, Co3, Ru, O'N, and He. For each group, the Lefschetz character is calculated, and its 2-modular block parts are examined for projectivity. In each case it is confirmed that a nonprincipal block part contains a nonprojective summand. The case of O'N is additionally found to have a nonprojective summand in its principal block part. Nineteen of the sporadic groups (including many previously known cases) are categorized into three classes based on projectivity properties of their Lefschetz characters. [ABSTRACT FROM AUTHOR]

Let A be a noetherian local k-algebra with residue field k and associated graded ring G(A). Let  be the conductor of A in its normalization [image omitted], which we assume to be regular and finite over A. Assume also that Proj(G(A)) is reduced and [image omitted]. Let r be the embedding dimension of A and [image omitted] be the Hilbert function of [image omitted]. We show that if the Hilbert function of Proj(G(A)) is equal to [image omitted], then G(A) is reduced. This allows us to compute, in the general case, the conductor of the local ring of singular isolated points on surfaces whose tangent cone is multiplanar; that is, it consists—as a set—of planes. We explicitly construct a class of such surfaces in the rational case. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0, and let A = k[x, y]/(f) be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e., a graded A-linear homomorphism ∇: Derk(A) → Endk(M), that satisfies the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring [image omitted] admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module. [ABSTRACT FROM AUTHOR]

A smooth algebraic surface S is said to be isogenous to a product of unmixed type if there exist two smooth curves C, F and a finite group G, acting faithfully on both C and F and freely on their product, so that S = (C × F)/G. In this article, we classify the surfaces of general type with pg = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian. It turns out that they belong to four families, that we call surfaces of type I, II, III, IV. The moduli spaces I, II, IV are irreducible, whereas III is the disjoint union of two irreducible components. In the last section we start the analysis of the case where G is not abelian, by constructing several examples. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ϵ cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type. [ABSTRACT FROM AUTHOR]