# Relations between simplicial groups, 3-crossed modules and 2-quadratic modules

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Let A, K, and H be subgroups of a group G and K â‰¤ H. Then we say that A covers the pair ( K, H) if AH = AK and avoids the pair ( K, H) if A âˆ© H = A âˆ© K. A pair ( K, H) in G is said to be maximal if K is a maximal subgroup of H. In the present paper, we study finite groups in...

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Let H be a weak Hopf algebra, and let A/B be a weak right H-Galois extension. In this paper, we mainly discuss the extension of the endomorphism algebra of a module over A. A necessary and sufficient condition for such an extension of the endomorphism algebra to be weak H-Galois is obtained by...

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Using simplicial groups, quasi 3-crossed modules of groups are intro- duced and some of the examples and results of quasi 3-crossed modules are given.

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In this paper, the definition of a Q-P quantale module and some relative concepts were introduced. Based on which, some properties of the Q-P quantale module, and the structure of the free Q-P quantale modules generated by a set were obtained. It was proved that the category of Q-P quantale...

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Let M be a right R-module. M is called X-H-supplemented if for every submodule A of M with A âˆŠ B(M;X) there exists a direct summand D of M such that M = A + Y if and only if M = D + Y for all Y âŠ† M. In this paper several properties of such modules are studied and we show that if M is...

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For a finite p-group P, the following three conditions are equivalent: (a) to have a (proper) partition, that is, to be the union of some proper subgroups with trivial pairwise intersections; (b) to have a proper subgroup all elements outside which have order p; (c) to be a semidirect product P...

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In this paper, we prove that the Hasse principle for any system of rational cubic forms, any system of rational homogeneous forms of degree at least 3 in an arbitrary number of variables is equivalent to the Hasse principle of certain systems of rational quadratic forms. This shows, in...

- A STRUCTURAL PROPERTY CONCERNING ABSTRACT COMMENSURABILITY OF SUBGROUPS. R. I. GRIGORCHUK; J. S. WILSON // Journal of the London Mathematical Society;Dec2003, Vol. 68 Issue 3, p671
Further properties of a group $\Gamma$ introduced by the first author in 1980 (sometimes called the first Grigorchuk group) are established. These are notably that any two infinite finitely generated subgroups of $\Gamma$ have isomorphic subgroups of finite index and that the generalized word...