On the Number of Fuzzy Subgroups and Fuzzy Normal Subgroups of Dihedral Group D8 and Quaternion group Q8

2021 ◽  
Vol 15 (3) ◽  
Keyword(s):  
Dihedral Group ◽  
Normal Subgroups ◽  
Quaternion Group ◽  
Fuzzy Subgroups

THE CONJUGACY CLASSES OF SOME FINITE METABELIAN GROUPS AND THEIR RELATED GRAPHS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Ain Asyikin Ibrahim ◽  
Alia Husna Mohd Noor ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Graph Theory ◽  
Conjugacy Class ◽  
Dihedral Group ◽  
Clique Number ◽  
Conjugacy Classes ◽  
Quaternion Group ◽  
Metabelian Groups ◽  
Dominating Number ◽  
Graph Properties ◽  
Class Graph

In this paper, the conjugacy classes of three metabelian groups, namely the Quasi-dihedral group, Dihedral group and Quaternion group of order 16 are computed. The obtained results are then applied to graph theory, more precisely to conjugate graph and conjugacy class graph. Some graph properties such as chromatic number, clique number, dominating number and independent number are found.   

scholarly journals Corrigenda On a classification of the function fields of algebraic tori: (Nagoya Math. J. 56 (1975), 85–104)

1980 ◽  
Vol 79 ◽  
pp. 187-190 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Dihedral Group ◽  
Quaternion Group ◽  
Algebraic Tori ◽  
Root Of Unity ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.

scholarly journals DETOUR ENERGY OF COMPLEMENT OF SUBGROUP GRAPH OF DIHEDRAL GROUP

2017 ◽  
Vol 1 (2) ◽  
Author(s):  
Abdussakir Abdussakir
Keyword(s):  
Dihedral Group ◽  
Normal Subgroups ◽  
Dihedral Groups ◽  
The Absolute

Study on the energy of a graph becomes a topic of great interest. One is the detour energy which is the sum of the absolute values of all eigenvalue of the detour matrix of a graph. Graphs obtained from a group also became a study that attracted the attention of many researchers. This article discusses the subgroup graph for several normal subgroups of dihedral groups. The discussion focused on the detour energy of complement of subgroup graph of dihedral group

An extension of a theorem of Kulakoff

1938 ◽  
Vol 34 (3) ◽  
pp. 316-320
Author(s):  
T. E. Easterfield
Keyword(s):  
Finite Order ◽  
Dihedral Group ◽  
Mixed Group ◽  
Quaternion Group ◽  
Sylow Subgroups ◽  
Cyclic Groups ◽  
Image Position

It has been shown by Kulakoff that if G is a group, not cyclic, of order pl, p being an odd prime, the number of subgroups of G of order pk, for 0 < k < l, is congruent to 1 + p (mod p2); and by Hall that if G is any group of finite order whose Sylow subgroups of G of order pk, p being odd, are not cyclic, then, for 0 < k < l, the number of subgroups of G of order pk is congruent to 1 + p (mod p2). No results were given for the case p = 2. In the present paper it is shown that analogous results hold for the case p = 2, but that the role of the cyclic groups is played by groups of four exceptional types: the cyclic groups themselves, and three non-Abelian types. These groups are defined as follows:(1) The dihedral group, of order 2k, generated by A and B, where(2) The quaternion group, of order 2k, generated by A and B, where(3) The "mixed" group, of order 2k, generated by A and B, where

scholarly journals UNITS IN F2D2p

2013 ◽  
Vol 13 (02) ◽  
pp. 1350090 ◽  
Author(s):  
KULDEEP KAUR ◽  
MANJU KHAN
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Dihedral Group ◽  
Unit Group ◽  
Normal Subgroups ◽  
Canonical Involution ◽  
Unitary Subgroup ◽  
Bicyclic Units

Let p be an odd prime, D2p be the dihedral group of order 2p, and F2 be the finite field with two elements. If * denotes the canonical involution of the group algebra F2D2p, then bicyclic units are unitary units. In this note, we investigate the structure of the group [Formula: see text], generated by the bicyclic units of the group algebra F2D2p. Further, we obtain the structure of the unit group [Formula: see text] and the unitary subgroup [Formula: see text], and we prove that both [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text].

scholarly journals Hurwitz Equivalence in Tuples of Generalized Quaternion Groups and Dihedral Groups

10.37236/804 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Xiang-dong Hou
Keyword(s):  
Dihedral Group ◽  
Quaternion Group ◽  
Dihedral Groups ◽  
Generalized Quaternion Group ◽  
Hurwitz Action ◽  
Generalized Quaternion

Let $Q_{2^m}$ be the generalized quaternion group of order $2^m$ and $D_N$ the dihedral group of order $2N$. We classify the orbits in $Q_{2^m}^n$ and $D_{p^m}^n$ ($p$ prime) under the Hurwitz action.

On non-abelian Brumer and Brumer–Stark conjecture for monomial CM-extensions

2014 ◽  
Vol 10 (04) ◽  
pp. 817-848 ◽  
Author(s):  
Jiro Nomura
Keyword(s):  
Dihedral Group ◽  
Number Fields ◽  
Alternating Group ◽  
Quaternion Group ◽  
Finite Set ◽  
Generalized Quaternion Group ◽  
Monomial Group ◽  
Quaternion Case ◽  
Stark Conjecture ◽  
Generalized Quaternion

Let K/k be a finite Galois CM-extension of number fields whose Galois group G is monomial and S a finite set of places of k. Then the "Stickelberger element" θK/k,S is defined. Concerning this element, Andreas Nickel formulated the non-abelian Brumer and Brumer–Stark conjectures and their "weak" versions. In this paper, when G is a monomial group, we prove that the weak non-abelian conjectures are reduced to the weak conjectures for abelian subextensions. We write D4p, Q2n+2 and A4 for the dihedral group of order 4p for any odd prime p, the generalized quaternion group of order 2n+2 for any natural number n and the alternating group on four letters respectively. Suppose that G is isomorphic to D4p, Q2n+2 or A4 × ℤ/2ℤ. Then we prove the l-parts of the weak non-abelian conjectures, where l = 2 in the quaternion case, and l is an arbitrary prime which does not split in ℚ(ζp) in the dihedral case and in ℚ(ζ3) in the alternating case. In particular, we do not exclude the 2-part of the conjectures and do not assume that S contains all finite places which ramify in K/k in contrast with Nickel's formulation.

scholarly journals ANTI SUBGRUP α-FUZZY

2020 ◽  
Vol 14 (1) ◽  
pp. 10
Author(s):  
Fiqriani Noor ◽  
Saman Abdurrahman ◽  
Naimah Hijriati
Keyword(s):  
Normal Subgroup ◽  
Group Structure ◽  
Normal Subgroups ◽  
Fuzzy Subset ◽  
New Concepts ◽  
Fuzzy Algebra ◽  
Sufficient And Necessary Conditions ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
The Relationship

The concept of fuzzy subgroups is a combination of the group structure with the fuzzy set, which was first introduced by Rosenfeld (1971). This concept became the basic concept in other the fuzzy algebra fields such as fuzzy normal subgroups, anti fuzzy subgroups and anti fuzzy normal subgroups. The development in the area of fuzzy algebra is characterized by the continual emergence of new concepts, one of which is the α-anti fuzzy subgroup concept. The idea of α-anti fuzzy subgroups is a combination between the α-anti fuzzy subset and anti fuzzy subgroups. The α-anti subset fuzzy which is an anti fuzzy subgroup is called as α-anti fuzzy subgroup. The purpose of this study is to prove that the α-anti fuzzy subset is an anti fuzzy subgroup, examine the relationship between α-anti fuzzy subgroups with anti fuzzy subgroups and α-fuzzy normal subgroups with anti fuzzy subgroups. The results of this study are, if A is an anti fuzzy subgroup (an anti fuzzy normal subgroup), then an α-anti subset fuzzy of A is an anti fuzzy subgroup (an anti fuzzy normal subgroup). However, this does not apply otherwise. Furthermore, this study also provides sufficient and necessary conditions for an α-anti fuzzy subset of any group to be an α-anti fuzzy subgroup and the formation of a group of factors that are built from an α-anti fuzzy normal subgroup.Keywords : Anti Fuzzy Subgroup, Anti Fuzzy Normal Subgroup, α-Anti Fuzzy Subgroup and α-Anti Fuzzy Normal Subgroup.

The lattices of fuzzy subgroups and fuzzy normal subgroups

1994 ◽  
Vol 76 (1-2) ◽  
pp. 1-11 ◽  
Author(s):  
Naseem Ajmal ◽  
K.V. Thomas
Keyword(s):  
Normal Subgroups ◽  
Fuzzy Subgroups
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