On quaternion-free 2-groups (original) (raw)

A characterization of the quaternion group

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2013

The goal of this note is to give an elementary characterization of the well-known quaternion group Q8 by using its subgroup lattice.

Finite groups with Quaternion Sylow subgroup

Comptes Rendus Mathématique, 2021

In this paper we show that a finite group G with Quaternion Sylow 2 -subgroup is 2 -nilpotent if, either 3 - |G| or G is solvable and the order of its Sylow 2-subgroup is strictly greater than 16

On infinite groups generated by two quaternions

Eprint Arxiv Math 0502512, 2005

Let x, y be two integral quaternions of norm p and l, respectively, where p, l are distinct odd prime numbers. We investigate the structure of x, y , the multiplicative group generated by x and y. Under a certain condition which excludes x, y from being free or abelian, we show for example that x, y , its center, commutator subgroup and abelianization are finitely presented infinite groups. We give many examples where our condition is satisfied and compute as an illustration a finite presentation of the group 1+j+k, 1+2j having these two generators and seven relations. In a second part, we study the basic question whether there exist commuting quaternions x and y for fixed p, l, using results on prime numbers of the form r 2 + ms 2 and a simple invariant for commutativity.

Some Properties of Representation of Quaternion Group

KnE Engineering

The quaternions are a number system in the form + + +. The quaternions ±1, ± , ± , ± form a non-abelian group of order eight called quaternion group. Quaternion group can be represented as a subgroup of the general linear group 2 (C). In this paper, we discuss some group properties of representation of quaternion group related to Hamiltonian group, solvable group, nilpotent group, and metacyclic group.

Complete decomposition of the generalized quaternion groups

Open Mathematics

Let G G be a finite nonabelian group. For any integer m ≥ 2 m\ge 2 , let A 1 , … , A m {A}_{1},\ldots ,{A}_{m} be nonempty subsets of G G . If A 1 , … , A m {A}_{1},\ldots ,{A}_{m} are mutually disjoint and if the subset product A 1 … A m = { α 1 … α m ∣ α v ∈ A v , v = 1 , 2 , … , m } {A}_{1}\ldots {A}_{m}=\left\{{\alpha }_{1}\ldots {\alpha }_{m}| {\alpha }_{v}\in {A}_{v},v=1,2,\ldots ,m\right\} coincides with G G , then ( A 1 , … , A m ) \left({A}_{1},\ldots ,{A}_{m}) is called a complete decomposition of G G of order m m . In this article, we let G G be the generalized quaternion groups Q 2 n {Q}_{{2}^{n}} , which is a finite nonabelian group of order 2 n {2}^{n} with group presentation given by ⟨ x , y ∣ x 2 n − 1 = 1 , y 2 = x 2 n − 2 , y x = x 2 n − 1 − 1 y ⟩ \langle x,y| {x}^{{2}^{n-1}}=1,{y}^{2}={x}^{{2}^{n-2}},yx={x}^{{2}^{n-1}-1}y\rangle for positive integer n ≥ 3 n\ge 3 . We determine the existence of complete decompositions of Q 2 n {Q}_{{2}^{n}} of order k k , for k ∈ {...

A class of nonabelian nonmetacyclic finite 2-groups

Glas Mat, 2006

Nonabelian nonmetacyclic finite 2-groups in which every proper subgroup is abelian or metacyclic and possessing at least one nonabelian and at least one nonmetacyclic proper subgroup have been investigated and classified. Using the obtained result and two previously known results one gets the complete classification of all nonabelian nonmetacyclic finite 2-groups in which every proper subgroup is abelian or metacyclic. 2000 Mathematics Subject Classification. 20D15.

On some metabelian 2-group and applications II

Boletim da Sociedade Paranaense de Matemática, 2016

Let G be some metabelian 2-group such that G/G ′ ≃ Z/2Z × Z/2Z × Z/2Z. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem of the 2-ideal classes of some fields k satisfying the condition Gal(k

A class of some third-metacyclic 2-groups

Novi Sad Journal of Mathematics, 2016

Third-metacyclic finite 2-groups are groups with a nonmetacyclic second-maximal subgroup and all its third-maximal subgroups being metacyclic. Among these groups we are looking for all of those whose non-metacyclic subgroups, including group itself, are generated by involutions.

On some metabelian 2-group whose abelianization is of type (2, 2, 2) and applications

Journal of Taibah University for Science

Let G be some metabelian 2-group satisfying the condition G/G ' is of type (2, 2, 2). In this paper, we construct all the sub- groups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we ap- ply these results to study the capitulation problem of the 2-ideal classes of some fi elds k satisfying the condition Gal(k^(2)_2/k) is of type (2, 2, 2), where k^(2)_2 is the second Hilbert 2-class eld of k.