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 Finite nilpotent groups that coincide with their 2-closures in all of their faithful permutation representations

2018 ◽  
Vol 17 (04) ◽  
pp. 1850065
Author(s):  
Alireza Abdollahi ◽  
Majid Arezoomand
Keyword(s):  
Nilpotent Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Nilpotent Groups ◽  
Quaternion Group ◽  
Finite Nilpotent Group ◽  
Closed Group ◽  
Odd Order ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

Let [Formula: see text] be any group and [Formula: see text] be a subgroup of [Formula: see text] for some set [Formula: see text]. The [Formula: see text]-closure of [Formula: see text] on [Formula: see text], denoted by [Formula: see text], is by definition, [Formula: see text] The group [Formula: see text] is called [Formula: see text]-closed on [Formula: see text] if [Formula: see text]. We say that a group [Formula: see text] is a totally[Formula: see text]-closed group if [Formula: see text] for any set [Formula: see text] such that [Formula: see text]. Here we show that the center of any finite totally 2-closed group is cyclic and a finite nilpotent group is totally 2-closed if and only if it is cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.

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.

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.

Classification of the affine structures of a generalized quaternion group of order ⩾32{\geqslant 32}

2020 ◽  
Vol 23 (5) ◽  
pp. 847-869
Author(s):  
Wolfgang Rump
Keyword(s):  
Adjoint Group ◽  
Quaternion Group ◽  
Isomorphism Classes ◽  
Affine Structures ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

AbstractBased on computing evidence, Guarnieri and Vendramin conjectured that, for a generalized quaternion group G of order {2^{n}\geqslant 32}, there are exactly seven isomorphism classes of braces with adjoint group G. The conjecture is proved in the paper.

scholarly journals Embedding Obstructions for the Generalized Quaternion Group

2000 ◽  
Vol 226 (1) ◽  
pp. 375-389 ◽  
Author(s):  
Ivo M. Michailov ◽  
Nikola P. Ziapkov
Keyword(s):  
Quaternion Group ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion
Sign in / Sign up

Export Citation Format

Share Document