scholarly journals On classical Krull dimension of group-graded rings

1997 ◽  
Vol 55 (2) ◽  
pp. 255-259 ◽  
Author(s):  
A. V. Kelarev

For any ring R graded by a finite group, we give a bound on the classical Krull dimension of R in terms of the dimension of the initial component Re. It follows that if Re has finite classical Krull dimension, then the same is true of the whole ring R, too.

2001 ◽  
Vol 131 (5) ◽  
pp. 1163-1166
Author(s):  
A. V. Kelarev

Giving as answer to Bergman's question, Cohen and Montgomery proved that, for every finite group G with identity e and each G-graded ring R = ⊕g∈GRg, the Jacobson radical J(Re) of the initial component Re is equal to Re ∩ J(R). We describe all semigroups S, which satisfy the following natural analogue of this property: J(Re) = Re ∩ J(R) for each S-graded ring R = ⊕s∈SRs and every idempotent e ∈ S.


2019 ◽  
Vol 12 (2) ◽  
pp. 332-347
Author(s):  
Najla SH. Al-Subaie ◽  
Mohammed Mosa Al-shomrani

The G-weak graded rings are rings graded by a set G of left coset representatives for the left action of a subgroup H of a finite group X. The main aim of this article is to study the concept of G-weak graded rings and continue the investigation of their properties. Moreover, some results concerning G-weak graded rings of fractions are derived. Finally, some additional examples of G-weak graded rings are provided.


Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.


2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.


2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.


Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .


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