• Title/Summary/Keyword: homological degree

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ON A FAMILY OF COHOMOLOGICAL DEGREES

  • Cuong, Doan Trung;Nam, Pham Hong
    • Journal of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.669-689
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    • 2020
  • Cohomological degrees (or extended degrees) were introduced by Doering, Gunston and Vasconcelos as measures for the complexity of structure of finitely generated modules over a Noetherian ring. Until now only very few examples of such functions have been known. Using a Cohen-Macaulay obstruction defined earlier, we construct an infinite family of cohomological degrees.

COMMUTING POWERS AND EXTERIOR DEGREE OF FINITE GROUPS

  • Niroomand, Peyman;Rezaei, Rashid;Russo, Francesco G.
    • Journal of the Korean Mathematical Society
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    • v.49 no.4
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    • pp.855-865
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    • 2012
  • Recently, we have introduced a group invariant, which is related to the number of elements $x$ and $y$ of a finite group $G$ such that $x{\wedge}y=1_{G{\wedge}G}$ in the exterior square $G{\wedge}G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m{\wedge}k$ of $H{\wedge}K$ such that $h^m{\wedge}k=1_{H{\wedge}K}$, where $m{\geq}1$ and $H$ and $K$ are arbitrary subgroups of $G$.

COMMUTING ELEMENTS WITH RESPECT TO THE OPERATOR Λ IN INFINITE GROUPS

  • Rezaei, Rashid;Russo, Francesco G.
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1353-1362
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    • 2016
  • Using the notion of complete nonabelian exterior square $G\hat{\wedge}G$ of a pro-p-group G (p prime), we develop the theory of the exterior degree $\hat{d}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs d(G). Among the main results of this paper, we describe upper and lower bounds for $\hat{d}(G)$ with respect to d(G). Here the size of the second homology group $H_2(G,\mathbb{Z}_p)$ (over the p-adic integers $\mathbb{Z}_p$) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of $H_2(G,\mathbb{Z}_p)$ both on G and $\hat{d}(G)$.