scholarly journals ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF FINITE MOUFANG LOOP

2021 ◽  
pp. 057
Author(s):  
Hamideh Hasanzadeh ◽  
Ali Iranmanesh ◽  
Behnam Azizi
Keyword(s):  
Finite Group ◽  
Dihedral Group ◽  
Moufang Loop ◽  
Infinite Class ◽  
Quaternion Group ◽  
Moufang Loops ◽  
A Finite Group

For a given element $g$ of a finite group $G$, the probablility that the commutator of randomly choosen pair elements in $G$ equals $g$ is the relative commutativity degree of $g$.  In this paper we are interested in studying the relative commutativity degree of the Dihedral group of order $2n$ and the Quaternion group of order $2^{n}$ for any $n\geq 3$ and we examine the relative commutativity degree of infinite class of the Moufang Loops of Chein type, $M(G,2)$.

The metric dimension of the enhanced power graph of a finite group

2019 ◽  
Vol 19 (01) ◽  
pp. 2050020 ◽  
Author(s):  
Xuanlong Ma ◽  
Yanhong She
Keyword(s):  
Finite Group ◽  
Dihedral Group ◽  
Metric Dimension ◽  
Quaternion Group ◽  
Power Graph ◽  
Vertex Set ◽  
Group A ◽  
Generalized Quaternion Group ◽  
A Finite Group ◽  
Generalized Quaternion

The enhanced power graph of a finite group [Formula: see text] is the graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if they generate a cyclic subgroup of [Formula: see text]. In this paper, we establish an explicit formula for the metric dimension of an enhanced power graph. As an application, we compute the metric dimension of the enhanced power graph of an elementary abelian [Formula: see text]-group, a dihedral group and a generalized quaternion group.

A Note on Special Local 2-Nilpotent Groups and the Solvability of Finite Groups

2018 ◽  
Vol 25 (04) ◽  
pp. 541-546
Author(s):  
Jiangtao Shi ◽  
Klavdija Kutnar ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Quaternion Group ◽  
A Finite Group

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, [Formula: see text] and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if [Formula: see text], and that GL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

The Coherence Number of 2-Groups

1990 ◽  
Vol 33 (4) ◽  
pp. 503-508 ◽  
Author(s):  
James McCool
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Dihedral Group ◽  
Upper Bounds ◽  
Distinguished Element ◽  
A Finite Group

AbstractLet G be a finite group. A natural invariant c(G) of G has been defined by W.J. Ralph, as the order (possibly infinite) of a distinguished element of a certain abelian group associated to G. Ralph has shown that c(Zn) = 1 and c(Z2 ⴲ Z2) = 2. In the present paper we show that c(G) is finite whenever G is a dihedral group or a 2-group, and obtain upper bounds for c(G) in these cases.

A Remark on the Units of Finite Order in The Group Ring of a Finite Group

1974 ◽  
Vol 17 (1) ◽  
pp. 129-130 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Finite Group ◽  
Finite Order ◽  
Direct Product ◽  
Group Ring ◽  
Integral Group Ring ◽  
Integral Group ◽  
Quaternion Group ◽  
Group Of Units ◽  
A Finite Group

Let G be a group, ZG its integral group ring and U(ZG) the group of units of ZG. The elements ±g∈U(ZG), g∈G, are called the trivial units of ZG. In this note we will proveLet G be a finite group. If ZG contains a non-trivial unit of finite order then it contains infinitely many non-trivial units of finite order.In [1] S. D. Berman has shown that if G is finite then every unit of finite order in ZG is trivial if and only if G is abelian or G is the direct product of a quaternion group of order 8 and an elementary abelian 2-group.

The metric dimension & distance spectrum of non-commuting graph of dihedral group

2021 ◽  
pp. 2150082
Author(s):  
Subarsha Banerjee
Keyword(s):  
Finite Group ◽  
Dihedral Group ◽  
Metric Dimension ◽  
Distance Spectrum ◽  
Commuting Graph ◽  
Vertex Set ◽  
A Finite Group

The non-commuting graph [Formula: see text] of a finite group [Formula: see text] has vertex set as [Formula: see text] and any two vertices [Formula: see text] are adjacent if [Formula: see text]. In this paper, we have determined the metric dimension and resolving polynomial of [Formula: see text], where [Formula: see text] is the dihedral group of order [Formula: see text]. The distance spectrum of [Formula: see text] has also been determined for all [Formula: see text].

Counting symmetric colorings of G × ℤ2

2019 ◽  
Vol 18 (10) ◽  
pp. 1950199
Author(s):  
Jabulani Phakathi ◽  
Shivani Singh ◽  
Yevhen Zelenyuk ◽  
Yuliya Zelenyuk
Keyword(s):  
Finite Group ◽  
Dihedral Group ◽  
A Finite Group

Let [Formula: see text] be a finite group and let [Formula: see text]. An [Formula: see text]-coloring of [Formula: see text] is any mapping [Formula: see text]. A coloring [Formula: see text] is symmetric if there is [Formula: see text] such that [Formula: see text] for every [Formula: see text]. We show that if [Formula: see text] is Abelian and [Formula: see text] is the polynomial representing the number of symmetric [Formula: see text]-colorings of [Formula: see text], then the number of symmetric [Formula: see text]-colorings of [Formula: see text] is [Formula: see text]. We also extend this result to the dihedral group [Formula: see text].

Lie automorphic loops under half-automorphisms

2019 ◽  
Vol 19 (11) ◽  
pp. 2050221 ◽  
Author(s):  
Maria De Lourdes Merlini Giuliani ◽  
Giliard Souza Dos Anjos
Keyword(s):  
Finite Group ◽  
Lie Ring ◽  
Multiplicative Systems ◽  
Odd Order ◽  
Moufang Loops ◽  
A Finite Group

Automorphic loops or [Formula: see text]-loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. Given a Lie ring [Formula: see text] we can define an operation [Formula: see text] such that [Formula: see text] is an [Formula: see text]-loop. We call it Lie automorphic loop. A half-isomorphism [Formula: see text] between multiplicative systems [Formula: see text] and [Formula: see text] is a bijection from [Formula: see text] onto [Formula: see text] such that [Formula: see text] for any [Formula: see text]. It was shown by [W. R. Scott, Half-homomorphisms of groups, Proc. Amer. Math. Soc. 8 (1957) 1141–1144] that if [Formula: see text] is a group then [Formula: see text] is either an isomorphism or an anti-isomorphism. This was used to prove that a finite group is determined by its group determinant. Here, we show that every half-automorphism of a Lie automorphic loop of odd order is either an automorphism or an anti-automorphism.

scholarly journals Total characters and Chebyshev polynomials

2003 ◽  
Vol 2003 (38) ◽  
pp. 2447-2453 ◽  
Author(s):  
Eirini Poimenidou ◽  
Homer Wolfe
Keyword(s):  
Finite Group ◽  
Chebyshev Polynomials ◽  
Irreducible Character ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Irreducible Characters ◽  
Integer Coefficients ◽  
Complete Answer ◽  
A Finite Group

The total characterτof a finite groupGis defined as the sum of all the irreducible characters ofG. K. W. Johnson asks when it is possible to expressτas a polynomial with integer coefficients in a single irreducible character. In this paper, we give a complete answer to Johnson's question for all finite dihedral groups. In particular, we show that, when such a polynomial exists, it is unique and it is the sum of certain Chebyshev polynomials of the first kind in any faithful irreducible character of the dihedral groupG.

A Lower Bound for the Real Genus of a Finite Group

1994 ◽  
Vol 46 (06) ◽  
pp. 1275-1286 ◽  
Author(s):  
Coy L. May
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Direct Product ◽  
Dihedral Group ◽  
Surface Group ◽  
Crystallographic Group ◽  
Klein Surface ◽  
Standard Representation ◽  
The Real ◽  
A Finite Group

Abstract Let G be a finite group. The real genus ρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we obtain a good general lower bound for the real genus of the group G. We use the standard representation of G as a quotient of a non-euclidean crystallographic group by a bordered surface group. The lower bound is used to determine the real genus of several infinite families of groups; the lower bound is attained for some of these families. Among the groups considered are the dicyclic groups and some abelian groups. We also obtain a formula for the real genus of the direct product of an elementary abelian 2-group and an “even” dicyclic group. In addition, we calculate the real genus of an abstract family of groups that includes some interesting 3-groups. Finally, we determine the real genus of the direct product of an elementary abelian 2-group and a dihedral group.

scholarly journals Some Results of The Coprime Graph of a Generalized Quaternion Group Q_4n

2021 ◽  
Vol 3 (1) ◽  
pp. 29-33
Author(s):  
Nurhabibah Nurhabibah ◽  
Abdul Gazir Syarifudin ◽  
I Gede Adhitya Wisnu Wardhana
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Quaternion Group ◽  
Group Order ◽  
Composite Number ◽  
Generalized Quaternion Group ◽  
A Finite Group ◽  
Generalized Quaternion

AbstractThe Coprime graph is a graph from a finite group that is defined based on the order of each element of the group. In this research, we determine the coprime graph of generalized quaternion group Q_(4n) and its properties. The method used is to study literature and analyze by finding patterns based on some examples. The first result of this research is the form of the coprime graph of a generalized quaternion group Q_(4n) when n = 2^k, n an odd prime number, n an odd composite number, and n an even composite number. The next result is that the total of a cycle contained in the coprime graph of a generalized quaternion group Q_(4n) and cycle multiplicity when  is an odd prime number is p-1.Keywords: Coprime graph, generalized quaternion group, order, path AbstrakGraf koprima merupakan graf dari dari suatu grup hingga yang didefiniskan berdasarkan orde dari masing-masing elemen grup tersebut. Pada penelitian ini akan dibahas tentang bentuk graf koprima dari grup generalized quaternion Q_(4n). Metode yang digunakan dalam penelitian ini adalah studi literatur dan melakukan analisis berdasarkan pola yang ditemukan dalam beberapa contoh. Adapun hasil pertama dari penelitian adalah bentuk graf koprima dari grup generalized quaternion Q_(4n) untuk kasus n = 2^k, n bilangan prima ganjil ganjil, n bilangan komposit ganjil dan n bilangan komposit genap. Hasil selanjutnya adalah total sikel pada graf koprima dari grup generalized quaternion dan multiplisitas sikel ketika  bilangan prima ganjil adalah p-1.Kata kunci: Graf koprima, grup generalized quternion, orde

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