# A Survey on the Estimation of Commutativity in Finite Groups

@inproceedings{Nath2013ASO, title={A Survey on the Estimation of Commutativity in Finite Groups}, author={Rajat Kanti Nath}, year={2013} }

Abstract. Let G be a finite group and let C = {(x, y) ∈ G × G ∣ xy = yx}. Then Pr(G) = ∣C∣/∣G∣ is the probability that two elements of G, chosen randomly with replacement, commute. This probability is a well known quantity, called commutativity degree of G, and indeed gives us an estimation of commutativity in G. In the last four decades this subject has enjoyed a flourishing development. In this article, we give a brief survey on the development of this subject and then we collect several of…

## 19 Citations

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Abstract We show some results on the probability that a randomly picked pair (H, K) of subgroups of a finite group G satisfies [H, K] = 1. This notion of probability is related with the subgroup…

### Autocommuting probability of a finite group

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ABSTRACT Let G be a finite group and Aut(G) the automorphism group of G. The autocommuting probability of G, denoted by Pr(G,Aut(G)), is the probability that a randomly chosen automorphism of G fixes…

### Common neighborhood spectrum of commuting graphs of finite groups

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The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and…

### Common Neighborhood Energy of Commuting Graphs of Finite Groups

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The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and…

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A finite non-abelian group $G$ is called super integral if the spectrum, Laplacian spectrum and signless Laplacian spectrum of its commuting graph contain only integers. In this paper, we first…

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Let $G$ be a finite non-abelian group with center $Z(G)$. The non-commuting graph of $G$ is a simple undirected graph whose vertex set is $Gsetminus Z(G)$ and two vertices $x$ and $y$ are adjacent…

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The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different subrings of a finite non-commutative ring equals a given…

### Laplacian Spectrum of Non-Commuting Graphs of Finite Groups

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In this paper, we compute the Laplacian spectrum of non-commuting graphs of some classes of finite non-abelian groups. Our computations reveal that the non-commuting graphs of all the groups…

### Laplacian Spectrum of Non-Commuting Graphs of Finite Groups

- MathematicsIndian Journal of Pure and Applied Mathematics
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In this paper, we compute the Laplacian spectrum of non-commuting graphs of some classes of finite non-abelian groups. Our computations reveal that the non-commuting graphs of all the groups…

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Let $R$ be a finite ring and $r \in R$. The aim of this paper is to study the probability that the commutator of a randomly chosen pair of elements of $R$ equals $r$.

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