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 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 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

On the Structure of Augmentation Quotient Groups for the Generalized Quaternion Group

2012 ◽  
Vol 19 (01) ◽  
pp. 137-148 ◽  
Author(s):  
Qingxia Zhou ◽  
Hong You
Keyword(s):  
Group Ring ◽  
Integral Group Ring ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Quaternion Group ◽  
Generalized Quaternion Group ◽  
Augmentation Quotient ◽  
Generalized Quaternion ◽  
Quotient Groups

For the generalized quaternion group G, this article deals with the problem of presenting the nth power Δn(G) of the augmentation ideal Δ (G) of the integral group ring ZG. The structure of Qn(G)=Δn(G)/Δn+1(G) is obtained.

SYMMETRY CLASSES OF POLYNOMIALS ASSOCIATED WITH THE DICYCLIC GROUP

2013 ◽  
Vol 06 (03) ◽  
pp. 1350033 ◽  
Author(s):  
Yousef Zamani ◽  
Esmaeil Babaei
Keyword(s):  
Symmetric Group ◽  
Quaternion Group ◽  
Irreducible Characters ◽  
Symmetry Classes ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

In this paper, we obtain the dimensions of symmetry classes of polynomials with respect to the irreducible characters of the dicyclic group as a subgroup of the full symmetric group. Then we discuss the existence of o-basis of these classes. In particular, the existence of o-basis of symmetry classes of polynomials with respect to the irreducible characters of the generalized quaternion group are concluded.

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.

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