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In this article we continue the study of R-factorizability in paratopological groups. It is shown that: (1) all concepts of R-factorizability in paratopological groups coincide; (2) a Tychonoff paratopological group G is R-factorizable if and only if it is totally ω -narrow and has property ω-QU; (3) every subgroup of a T1 paratopological group G is R-factorizable provided that the topological group G? associated to G is a Lindelöf Σ-space, i.e., G is a totally Lindelöf Σ-space ; (4) if Π=∏i∈IGi is a product of T1 paratopological groups which are totally Lindelöf Σ-spaces, then each dense subgroup of Π is R-factorizable. These results answer in the affirmative several questions posed earlier by M. Sanchis and M. Tkachenko and by S. Lin and L.-H. Xie. 相似文献
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Let k be any field, G be a finite group acting on the rational function field k(xg:g∈G) by h⋅xg=xhg for any h,g∈G. Define k(G)=k(xg:g∈G)G. Noether’s problem asks whether k(G) is rational (= purely transcendental) over k. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether’s problem. We prove that, if G is a Frobenius group with abelian Frobenius kernel, then k(G) is retract k-rational for any field k satisfying some mild conditions. As an application, we show that, for any algebraic number field k, for any Frobenius group G with Frobenius complement isomorphic to SL2(F5), there is a Galois extension field K over k whose Galois group is isomorphic to G, i.e. the inverse Galois problem is valid for the pair (G,k). The same result is true for any non-solvable Frobenius group if k(ζ8) is a cyclic extension of k. 相似文献
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We consider the p -Zassenhaus filtration (Gn) of a profinite group G . Suppose that G=S/N for a free profinite group S and a normal subgroup N of S contained in Sn. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p -cohomological dimension ≤ 1), we prove that Gn+1 is the intersection of all kernels of upper-triangular unipotent (n+1)-dimensional representations of G over Fp. This extends earlier results by Miná?, Spira, and the author on the structure of absolute Galois groups of fields. 相似文献