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Staic defined symmetric cohomology of groups and studied that the secondary symmetric cohomology group is corresponding to group extensions and the injectivity of the canonical map from symmetric cohomology to classical cohomology. In this paper, we define symmetric cohomology and symmetric Hochschild cohomology for cocommutative Hopf algebras. The first one is a generalization of symmetric cohomology of groups. We give an isomorphism between symmetric cohomology and symmetric Hochschild cohomology, which is a symmetric version of the classical result about cohomology of groups by Eilenberg and MacLane and cohomology of Hopf algebras by Ginzburg and Kumar. Moreover, to consider the condition that symmetric cohomology coincides with classical cohomology, we investigate the projectivity of a resolution which gives symmetric cohomology. [ABSTRACT FROM AUTHOR]

We prove a Poincaré duality theorem for finite groups, where the (co)homologies involved are a variant of classical group cohomology due to Zarelua and Staic, denoted by H λ ⁎. In particular, we show that in certain cases (but not always), for a finite group G of order n and a G -module M , we have the isomorphism H λ n − i − 1 (G , M) → ≅ H i λ (G , M tw) , where M tw is a twisting of M defined in Section 5. [ABSTRACT FROM AUTHOR]

The paper investigates exterior and symmetric (co)homologies of groups. We introduce symmetric homology of groups and compute exterior and symmetric (co)homologies of some finite groups. We also compare the classical, exterior and symmetric (co)homologies. Finally, we derive restriction and corestriction homomorphisms for exterior cohomology. [ABSTRACT FROM AUTHOR]

This paper links the third symmetric cohomology (introduced by Staic [10] and Zarelua [12]) to crossed modules with certain properties. The equivalent result in the language of 2- groups states that an extension of 2-groups corresponds to an element of HS³ iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding HS² and abelian extensions of groups. [ABSTRACT FROM AUTHOR]

The symmetric cohomology group of a commutative semigroup is defined in dimension 4 and is shown to equal its triple cohomology group. [ABSTRACT FROM AUTHOR]

Symmetric cohomology of groups, defined by M. Staic in [2], is similar to the way one defines the cyclic cohomology for algebras. We show that there is a welldefined restriction, conjugation and transfer map in symmetric cohomology, which form a Mackey functor under a restriction. Some new properties for the symmetric cohomology group using normalized cochains are also given. [ABSTRACT FROM AUTHOR]

If Γ is a group, then braided Γ-crossed modules are classified by braided strict Γ-graded categorial groups. The Schreier theory obtained for Γ-module extensions of the type of an abelian Γ-crossed module is a generalization of the theory of Γ-module extensions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a symmetric continuous cohomology of topological groups. This is obtained by topologizing a recent construction due to Staic [23], where a symmetric cohomology of abstract groups is constructed. We give a characterization of topological group extensions that correspond to elements of the second symmetric continuous cohomology. We also show that the symmetric continuous cohomology of a profinite group with coefficients in a discrete module is equal to the direct limit of the symmetric cohomology of finite groups. In the end, we also define symmetric smooth cohomology of Lie groups and prove similar results. [ABSTRACT FROM AUTHOR]

In this paper, we state the notion of morphisms in the category of abelian crossed modules and prove that this category is equivalent to the category of strict Picard categories and regular symmetric monoidal functors. The theory of obstructions for symmetric monoidal functors and symmetric cohomology groups are applied to show a treatment of the group extension problem of the type of an abelian crossed module. [ABSTRACT FROM AUTHOR]

Abstract: We introduce Δ-groups and show how they fit in the context of lattice field theory. To a topological space M we associate a Δ-group . We define the symmetric cohomology of a group G with coefficients in a G-module A. The Δ-group is determined by the action of on and an element of . [Copyright &y& Elsevier]