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On quaternion-free 2-groups

Bettina Wilkens
2002 Journal of Algebra  
Two theorems are proved, the first 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.  ... 
doi:10.1016/s0021-8693(02)00638-5 fatcat:4k6r7mvq6rfa7nbb3x3pgmnw2e

Free groups in quaternion algebras

S.O. Juriaans, A.C. Souza Filho
2013 Journal of Algebra  
In [8] we constructed pairs 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}.  ... 
doi:10.1016/j.jalgebra.2012.12.025 fatcat:n4gxrz5vbjfgbeuxaxnuxpxjry

Schur Indices in Finite Quaternion-Free Groups

A. D. Oh
1982 Proceedings of the American Mathematical Society  
Let G be a finite, 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.  ... 
doi:10.2307/2044054 fatcat:bh2bhhfzsnbuhjy2hrwsdargai

Schur indices in finite quaternion-free groups

A. D. Oh
1982 Proceedings of the American Mathematical Society  
Let G be a finite, 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.  ... 
doi:10.1090/s0002-9939-1982-0660593-5 fatcat:twwbe2xt2zaxxnnmmdng2iptbu

Examples of exotic free 2–complexes and stably free nonfree modules for quaternion groups

F Rudolf Beyl, Nancy Waller
2008 Algebraic and Geometric Topology  
This is a continuation 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  ... 
doi:10.2140/agt.2008.8.1 fatcat:4qsvfjk5g5bk5j6fk42bik7fne

POLYCYCLIC PRESENTATIONS OF THE TORSION FREE SPACE GROUP WITH QUATERNION POINT GROUP OF ORDER EIGHT

Siti Afiqah Mohammad, Nor Haniza Sarmin, Hazzirah Izzati Mat Hassim
2015 Jurnal Teknologi  
Therefore, this research focuses on computing the polycyclic presentations 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.  ... 
doi:10.11113/jt.v77.7020 fatcat:ozoladoqgjf4ffs73r5w32if34

On finite groups whose Sylow subgroups are modular or quaternion-free

Gary M Seitz, C.R.B Wright
1969 Journal of Algebra  
Since the spmmetric 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.  ... 
doi:10.1016/0021-8693(69)90080-5 fatcat:ktcq7zfogfhytccrfopiq3befa

A note on modular p-groups

Robert W van der Waall
1973 Indagationes Mathematicae (Proceedings)  
[A section 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.  ... 
doi:10.1016/1385-7258(73)90021-8 fatcat:o2y4v2hmxfaa5ogxj2beyj3f6a

A Note on p-Nilpotence of Finite Groups

Wujie Shi
2001 Journal of Algebra  
Thus P s a, b is a Sylow 2-subgroup 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  ... 
doi:10.1006/jabr.2001.8748 fatcat:mf4c5olxoje5lp5ajb2acw4hei

The hypercomplex quotient and the quaternionic quotient

Dominic Joyce
1991 Mathematische Annalen  
We note that in the classification by Berger [B] 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).  ... 
doi:10.1007/bf01459248 fatcat:s5ijbcaocvacnmdoc3a63hykb4

A note on p-nilpotence and solvability of finite groups

Jiangtao Shi, Wujie Shi, Cui Zhang
2009 Journal of Algebra  
Let M be a nilpotent maximal subgroup 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.  ... 
doi:10.1016/j.jalgebra.2008.12.004 fatcat:jmfartjotreepbguxhpdsjehl4

Free Topological Groups

B. R. Gelbaum
1961 Proceedings of the American Mathematical Society  
Since [4; 5; 8] the 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.  ... 
doi:10.2307/2034868 fatcat:itu5fwdtyrdu3c36lml6674wty

Free topological groups

B. R. Gelbaum
1961 Proceedings of the American Mathematical Society  
Since [4; 5; 8] the 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.  ... 
doi:10.1090/s0002-9939-1961-0140607-8 fatcat:5jlbwsylnzclbeqgl7rp2eb63e

Free subgroups of quaternion algebras

Roger C. Alperin
1993 Proceedings of the American Mathematical Society  
Using the theory 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.  ... 
doi:10.1090/s0002-9939-1993-1123646-0 fatcat:d5rdfneu2raqbogu67hit3tuwe

Free Subgroups of Quaternion Algebras

Roger C. Alperin
1993 Proceedings of the American Mathematical Society  
Using the theory 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.  ... 
doi:10.2307/2160003 fatcat:sczsa74n5befrlwkayluez4fmq
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