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On quaternion-free 2-groups

2002
*
Journal of Algebra
*

Two theorems are proved, the first

doi:10.1016/s0021-8693(02)00638-5
fatcat:4k6r7mvq6rfa7nbb3x3pgmnw2e
*of*them showing that a modular*quaternion*-*free*finite 2-*group*has a characteristic abelian subgroup with metacyclic factor, the second classifying nonmodular finite*quaternion*-*free*... 2-*groups*. ... This says that H /A is either cyclic or generalized*quaternion*; so H /A is cyclic, H being Q 8 -*free*. ✷ Lemma 11. Let P be a finite Q 8 -*free*2-*group*. ...##
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Free groups in quaternion algebras

2013
*
Journal of Algebra
*

In [8] we constructed pairs

doi:10.1016/j.jalgebra.2012.12.025
fatcat:n4gxrz5vbjfgbeuxaxnuxpxjry
*of*units u, v in Z-orders*of*a*quaternion*algebra over Q( √ −d), d ≡ 7 (mod 8) positive and square*free*, such that u n , v n is*free*for some n ∈ N. ... not applied to show whether the*group*they generate is*free*or not. * Mathematics Subject Classification Primary [16U 60, 20E05]; Secondary [16S34, 20M 05]. ...*Free**Groups*in*Quaternion*Algebras In the sequel, K = Q( √ −d) is an imaginary quadratic extension with d a positive and square-*free*integer. Let ξ = ψ be elements*of*{1, i, j, k}. ...##
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Schur Indices in Finite Quaternion-Free Groups

1982
*
Proceedings of the American Mathematical Society
*

Let G be a finite,

doi:10.2307/2044054
fatcat:bh2bhhfzsnbuhjy2hrwsdargai
*quaternion*-*free**group*with exponent e, let F be a field*of*characteristic zero and let x be an absolutely irreducible character*of*G. ... Suppose that a Sylow 2-subgroup*of*the Galois*group**of*F(J\) over F is cyclic. It is shown that if x is not real valued, then the Schur index*of*x over F is odd. ... Following [6] we call G a*quaternion*-*free**group*if a generalized*quaternion**group*is not involved in G. In [3] , B. ...##
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Schur indices in finite quaternion-free groups

1982
*
Proceedings of the American Mathematical Society
*

Let G be a finite,

doi:10.1090/s0002-9939-1982-0660593-5
fatcat:twwbe2xt2zaxxnnmmdng2iptbu
*quaternion*-*free**group*with exponent e, let F be a field*of*characteristic zero and let x be an absolutely irreducible character*of*G. ... Suppose that a Sylow 2-subgroup*of*the Galois*group**of*F(J\) over F is cyclic. It is shown that if x is not real valued, then the Schur index*of*x over F is odd. ... Following [6] we call G a*quaternion*-*free**group*if a generalized*quaternion**group*is not involved in G. In [3] , B. ...##
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Examples of exotic free 2–complexes and stably free nonfree modules for quaternion groups

2008
*
Algebraic and Geometric Topology
*

This is a continuation

doi:10.2140/agt.2008.8.1
fatcat:4qsvfjk5g5bk5j6fk42bik7fne
*of*our study [3]*of*a family*of*projective modules over Q 4n , the generalized*quaternion*(binary dihedral)*group**of*order 4n. Our approach is constructive. ... This paper offers an infinite collection*of*finite*groups*with stably*free*nonfree modules P , given as ideals in the*group*ring. ... A class*of*projective modules over ZQ 4 n Here the underlying*group*is the generalized*quaternion*(binary dihedral)*group*Q 4n D hx; yˇx n D y 2 D .xy/ 2 i ; n 2 : Given integers a and b , not both zero ...##
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POLYCYCLIC PRESENTATIONS OF THE TORSION FREE SPACE GROUP WITH QUATERNION POINT GROUP OF ORDER EIGHT

2015
*
Jurnal Teknologi
*

Therefore, this research focuses on computing the polycyclic presentations

doi:10.11113/jt.v77.7020
fatcat:ozoladoqgjf4ffs73r5w32if34
*of*the torsion*free*space*group*named Bieberbach*group*with a*quaternion*point*group**of*order eight. ... One*of*the properties is on exploration*of*the nonabelian tensor square*of*the*group*. ... In this paper, our focus is on the torsion*free*space*group*with*quaternion*point*group**of*order eight. ...##
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On finite groups whose Sylow subgroups are modular or quaternion-free

1969
*
Journal of Algebra
*

Since the spmmetric

doi:10.1016/0021-8693(69)90080-5
fatcat:ktcq7zfogfhytccrfopiq3befa
*group*S, ha.s*quaternion*-*free*2-Sylow subgroup and 2-length 2, we cannot conclude imrncdiatciy that G* has 2-length 1 whenever it has*quaternion*-*free*2-Sylow subgroups. ... ey non-abelian*quaternion*-*free*2-*group*has a characteristic maximal subgroup. One*of*the main results in this paper is an analogous result for modular p-*groups*. THEOREM 1. ...##
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A note on modular p-groups

1973
*
Indagationes Mathematicae (Proceedings)
*

[A section

doi:10.1016/1385-7258(73)90021-8
fatcat:o2y4v2hmxfaa5ogxj2beyj3f6a
*of*a*group*G is by definition a homomorphic image*of*a subgroup*of*G]. A 2-*group*is called*quaternion*-*free*if it has no section isomorphic to the*quaternion**group*&*of*eight elements. ... It is the purpose*of*this paper to derive a criterion for a non-abelian p-*group*to be modular (*quaternion*-*free*when p= 2). Note that all abelian p-*groups*are modular and*quaternion*-*free*as well. ...##
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A Note on p-Nilpotence of Finite Groups

2001
*
Journal of Algebra
*

Thus P s a, b is a Sylow 2-subgroup

doi:10.1006/jabr.2001.8748
fatcat:mf4c5olxoje5lp5ajb2acw4hei
*of*GL 2, 3 and is a semidihedral ² 2 :*group**of*order 16. Also Q s a , ab is a*quaternion**group**of*order 8 and G is not*quaternion*-*free*. ... If P is*quaternion*-*free*and N P is 1 G 2-nilpotent, then G is 2-nilpotent. w x Then in 1 they state they ''do not know any examples*of**groups*which show that the*quaternion*-*free*hypothesis is necessary ...##
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The hypercomplex quotient and the quaternionic quotient

1991
*
Mathematische Annalen
*

We note that in the classification by Berger [B]

doi:10.1007/bf01459248
fatcat:s5ijbcaocvacnmdoc3a63hykb4
*of*holonomy*groups**of*manifolds with torsion-*free*connections, SL(n,I-IJU(I) is given as a possible holonomy*group*in Theorem 4, p. 320; in Berger's notation ... Now a*quaternionic*structure on lEF 2, 1.k will be given as a*quaternionic*quotient*of*I-I~ 2 by the*group*U(I). ...##
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A note on p-nilpotence and solvability of finite groups

2009
*
Journal of Algebra
*

Let M be a nilpotent maximal subgroup

doi:10.1016/j.jalgebra.2008.12.004
fatcat:jmfartjotreepbguxhpdsjehl4
*of*a finite*group*G, and P a Sylow 2-subgroup*of*M. IfP is*quaternion*-*free*and Ω 1 but G is not solvable. ... In [1] the authors remark that they do not know any examples*of**groups*which show that the*quaternion*-*free*hypothesis is necessary in Theorem 2. In [2] W. ...##
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Free Topological Groups

1961
*
Proceedings of the American Mathematical Society
*

Since [4; 5; 8] the

doi:10.2307/2034868
fatcat:itu5fwdtyrdu3c36lml6674wty
*group**of*rotations in 3-space contains a*free**group*having infinitely many generators Ri, R2, ■ ■ ■ , the corresponding*quaternions*q\, q2, ■ ■ • form a system*of*generators for a*free*...*Free*topological*groups*. ...##
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Free topological groups

1961
*
Proceedings of the American Mathematical Society
*

Since [4; 5; 8] the

doi:10.1090/s0002-9939-1961-0140607-8
fatcat:5jlbwsylnzclbeqgl7rp2eb63e
*group**of*rotations in 3-space contains a*free**group*having infinitely many generators Ri, R2, ■ ■ ■ , the corresponding*quaternions*q\, q2, ■ ■ • form a system*of*generators for a*free*...*Free*topological*groups*. ...##
###
Free subgroups of quaternion algebras

1993
*
Proceedings of the American Mathematical Society
*

Using the theory

doi:10.1090/s0002-9939-1993-1123646-0
fatcat:d5rdfneu2raqbogu67hit3tuwe
*of**group*actions on trees, we shall prove that if a*quaternion*algebra over Laurant polynomials is not split then a certain congruence subgroup*of*the*group**of*norm one elements is a*free*...*group*. ... is a nonsplit*quaternion*algebra then T is a*free**group*. Proof. ...##
###
Free Subgroups of Quaternion Algebras

1993
*
Proceedings of the American Mathematical Society
*

Using the theory

doi:10.2307/2160003
fatcat:sczsa74n5befrlwkayluez4fmq
*of**group*actions on trees, we shall prove that if a*quaternion*algebra over Laurant polynomials is not split then a certain congruence subgroup*of*the*group**of*norm one elements is a*free*...*group*. ... is a nonsplit*quaternion*algebra then T is a*free**group*. Proof. ...
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