• Corpus ID: 208264455

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… 
math.ipm.ac.ir
19 Citations
Highly Influential Citations
2
Background Citations
10
Methods Citations
1

Tables from this paper

19 Citations

Strong subgroup commutativity degree and some recent problems on the commuting probabilities of elements and subgroups

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

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

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

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…

On super integral groups

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…

On Laplacian energy of non-commuting graphs of finite groups

‎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…

On commuting probability of finite rings II

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

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

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…

On generalized commuting probability of finite rings

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 $r$.

References

SHOWING 1-10 OF 38 REFERENCES

On the Relative Commutativity Degree of a Subgroup of a Finite Group

The aim of this article is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of relative commutativity degree of a subgroup H of a…

ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP

Given any two subgroups H and K of a finite group G, and an element g ∈ G, the aim of this article is to study the probability that the commutator of an arbitrarily chosen pair of elements (one from…

Commutativity in non-Abelian Groups

Let P2(G) be defined as the probability that any two elements selected at random from the group G, commute with one another. If G is an Abelian group, P2(G) = 1, so our interest lies in the…

On Solutions of a Class of Equations in a Finite Group

Let G be a finite group and ω(x 1, x 2,…, x n ) denote the product of x 1, x 2,…, x n , in a randomly chosen order. The object of this article is to prove that the number of solutions of the equation…

Probability that the commutator of two group elements is equal to a given element

On a lower bound of commutativity degree

The commutativity degree of a finite group G, denoted by Pr(G), is the probability that a randomly chosen pair of elements of G commute. The object of this paper is to derive a lower bound for Pr(G)…

Commutators and commutator subgroups

A Generalization of Commutativity Degree of Finite Groups

The commutativity degree of a finite group is the probability that two arbitrarily chosen group elements commute. This notion has been generalized in a number of ways. The object of this article is…

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

Abstract We compute commutativity degrees of wreath products $A \wr B$ of finite Abelian groups A and B. When B is fixed of order n the asymptotic commutativity degree of such wreath products is…

Character Theory of Finite Groups

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is…