Abstract
Letm, n be positive integers. We denote byR(m, n) (respectivelyP(m, n)) the class of all groupsG such that, for everyn subsetsX 1, X2, . . .,X n of sizem ofG there exits a non-identity permutation σ such that\(X_1 X_2 ...X_n \cap X_{\sigma (1)} X_{\sigma (2)} ...X_{\sigma (n)} \ne \not 0\) (respectively X1X2 . . .X n = Xσ(1)X{σ(2)} . . . X{gs(n)}). Let G be a non-abelian group. In this paper we prove that
-
(i)
G ∈ P(2,3) if and only ifG isomorphic to S3, whereS n is the symmetric group onn letters.
-
(ii)
G ∈ R(2, 2) if and only if¦G¦ ≤ 8.
-
(iii)
IfG is finite, thenG ∈ R(3, 2) if and only if¦G¦ ≤ 14 orG is isomorphic to one of the following: SmallGroup(16,i), i ∈ {3, 4, 6, 11, 12, 13}, SmallGroup(32,49), SmallGroup(32, 50), where SmallGroup(m, n) is the nth group of orderm in the GAP [13] library.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Abdollahi and A. Mohammadi Hassanabadi,A charactrization of infinite abelian groups, Bull. Iranian Math. Soc.24(2) (1998), 42–47.
A. Abdollahi, A. Mohammadi Hassanabadi and B. Taeri,A property equivalent to n- permutability for infinite groups, J. Algebra221 (1999), 570–578.
R. D. Blyth,Rewriting products of group elements I, J. Algebra116 (1988), 506–521.
M. Curzio, P. Longobardi and M. Maj,Su di problema combinatoria in teoria dei gruppi, Atti Acc Lincei Rend fis. VIII,LXXIV (1983) 136–142.
M. Curzio, P. Longobardi, M. Maj and D. J. S. Robinson,A permutational property of groups, Arch. Math. (Basel)44 (1985), 385–389.
P. Longobardi, M. Maj and A. H. Rhemtulla,Infinite groups in a given variety and Ram- sey’s Theorem, Comm. Algebra20 (1992), 127–139.
P. Longobardi, M. Maj and S. E. Stonehewer,The classification of groups in which every product of four elements can be reordered, Rend. Sem. mat. Univ. Padova93 (1995), 253–261.
P. Longobardi, M. Maj and S. E. Stonehewer,Finite 2-groups in which every product of four elements can be reordered, Illinois Journal of Mathematics35 (1991), 198–214.
M. Maj and S. E. Stonehewer,Non-nilpotent groups in which every product of four ele- ments can be reordered, Canad. J. Math.XLII No. 6, (1990), 1053–1066.
A. Mohammadi Hassanabadi,A property equivalent to permutablity for groups, Rend. Sem. Mat. Univ. Padova100 (1998), 137–142.
A. Mohammadi Hassanabadi and A. H. Rhemtulla,criteria commutativity in large groups, Canad. Math. Bull.41(1) (1998), 65–70.
D. J. S. Robinson,A course in the theory of groups, 2nd ed., Springer-Verlag, Berlin, 1996.
M. Schonert et al.,GAP: Groups, algorithms, and programming, Lehrstuhl D fur Mathe- matik, RWTH Aachen, 2002.
Author information
Authors and Affiliations
Additional information
The author was supported in part by Isfahan University of Technology, no. 1MAC811.
Rights and permissions
About this article
Cite this article
Taeri, B. On a permutablity problem for groups. J. Appl. Math. Comput. 20, 75–96 (2006). https://doi.org/10.1007/BF02831925
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02831925