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This paper serves as a source of examples of Rokhlin actions or locally representable actions of finite groups on C*-algebras satisfying a certain UHF-absorption condition. We show that given any finite group G and a separable, unital C*-algebra A that absorbs M|G|∞ tensorially, one can lift any group homomorphism G → Aut(A)/≈u to an honest Rokhlin action γ of G on A. Unitality may be dropped in favour of stable rank one or being stable. If A belongs to a certain class of C*-algebras that is classifiable by a suitable invariant (e.g. K-theory), then in fact every G-action on the invariant lifts to a Rokhlin action of G on A. For the crossed product C*-algebra A ⋊γ G of a Rokhlin action on a UHF-absorbing C*-algebra, an inductive limit decomposition is obtained in terms of A and γ. If G is assumed to be abelian, then the dual action ˆγ is locally representable in a very strong sense. We then show how some well-known constructions of finite group actions with certain predescribed properties can be recovered and extended by the main results of this paper when paired with known classification theorems. Among these is Blackadar's famous construction of symmetries on the CAR algebra whose fixed point algebras have non-trivial K1-groups. Lastly, we use the results of this paper to reduce the UCT problem for separable, nuclear C*-algebras to a question about certain finite group actions on O2. [ABSTRACT FROM AUTHOR]

In this paper we introduce new densities on the set of primes of a number field. If K/K0 is a Galois extension of number fields, we associate to any element x ∈ ... a density δK/K0,x on the primes of K. In particular, the density associated to x = 1 is the usual Dirichlet density on K. We also give two applications of these densities (for x ≠ 1): the first is a realization result à la the Grunwald-Wang theorem such that essentially, ramification is only allowed in a set of arbitrarily small (positive) Dirichlet density. The second concerns the so-called saturated sets of primes, introduced by Wingberg. [ABSTRACT FROM AUTHOR]

For a finite group G and an arbitrary commutative ring R, Broué has placed a Frobenius exact structure on the category of finitely generated RG-modules by taking the exact sequences to be those that split upon restriction to the trivial subgroup. The corresponding stable category is then tensor triangulated. In this paper we examine the case R = S/tn, where S is a discrete valuation ring having uniformising parameter t. We prove that the prime ideal spectrum (in the sense of Balmer) of this 'relative' version of the stable module category of RG is a disjoint union of n copies of that for kG, where k is the residue field of S. [ABSTRACT FROM AUTHOR]

We correct two errors in the statement and proof of a theorem in an earlier paper (2007), and at the same time extend that result to a more general theorem about extensions of $p$-local finite groups. Other special cases of this theorem have already been shown in two later papers, so we feel it will be useful to have this more general result in the literature. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic p &#62 0,and let G be a simple, simply connected algebraic group defined over Fp.Given r ≥ 1, set q = pr, and let G(Fq) be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group H1(G(Fq), L(λ)), where L(λ) is the simple G-module of highest weight λ. Under certain very mild conditions on p and q, we are able to completely describe the first cohomology group when λ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when λ is a minimal non-zero dominant weight. [ABSTRACT FROM AUTHOR]

In this paper, we compute the Gerstenhaber bracket on the Hoch-schild cohomology of $C^infty (M)rtimes G$ for a finite group $G$ acting on a compact manifold $M$. Using this computation, we obtain geometric descriptions for all noncommutative Poisson structures on $C^infty (M)rtimes G$ when $M$ is a symplectic manifold. We also discuss examples of deformation quantizations of these noncommutative Poisson structures. [ABSTRACT FROM AUTHOR]

The Ito-Michler theorem asserts that if no irreducible character of a finite group $G$ has degree divisible by some given prime $p$, then a Sylow $p$-subgroup of $G$ is both normal and abelian. In this paper we relax the hypothesis, and we assume that there is at exactly one multiple of $p$ that occurs as the degree of an irreducible character of $G$. We show that in this situation, a Sylow $p$-subgroup of $G$ is almost normal in $G$, and it is almost abelian. [ABSTRACT FROM AUTHOR]

We introduce new biholomorphic invariants for real-analytic hypersurfaces in $mathbb {C}^2$ and show how they can be used to show that a hypersurface possesses few automorphisms. We give conditions, in terms of the new invariants, guaranteeing that the stability group is finite, and give (sharp) bounds on the cardinality of the stability group in this case. We also give a sufficient condition for the stability group to be trivial. The main technical tool developed in this paper is a complete (formal) normal form for a certain class of hypersurfaces. As a byproduct, a complete classification, up to biholomorphic equivalence, of the finite type hypersurfaces in this class is obtained. [ABSTRACT FROM AUTHOR]

Let $G$ be a finite group, $S in mathit {Syl}_p(G)$, and $mathcal S$ be the set subgroups containing $S$. For $M in mathcal S$ and $V = Omega _1Z(O_p(M))$, the paper discusses the action of $M$ on $V$. Apart from other results, it is shown that for groups of parabolic characteristic $p$ either $S$ is contained in a unique maximal $p$-local subgroup, or there exists a maximal $p$-local subgroup in $M in mathcal S$ such that $V$ is a nearly quadratic 2F-module for $M$. [ABSTRACT FROM AUTHOR]