×

On matchable subsets in abelian groups and their linear analogues. (English) Zbl 1426.05174

Summary: In this paper, we introduce the notions of matching matrices in groups and vector spaces, which lead to some necessary conditions for existence of acyclic matching in abelian groups and its linear analogue. We also study the linear local matching property in field extensions to find a dimension criterion for linear locally matchable bases. Moreover, we define the weakly locally matchable subspaces and we investigate their relations with matchable subspaces. We provide an upper bound for the dimension of primitive subspaces in a separable field extension. We employ MATLAB coding to investigate the existence of acyclic matchings in finite cyclic groups. Finally, a possible research problem on matchings in \(n\)-groups is presented. Our tools in this paper mix combinatorics and linear algebra.

MSC:

05D15 Transversal (matching) theory
11B75 Other combinatorial number theory
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F99 Special aspects of infinite or finite groups
12F99 Field extensions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aliabadi, M.; Hadian, M.; Jafari, A., On matching property for groups and field extensions, J. Algebra Appl., 15, 1, Article 1650011 pp. (2016) · Zbl 1330.05158
[2] Aliabadi, M.; Janardhanan, M. J., On local matching property in groups and vector spaces, Australas. J. Combin., 70, 75-85 (2018) · Zbl 1441.05222
[3] Alon, N.; Fan, C. K.; Kleitman, D.; Losonczy, J., Acyclic matchings, Adv. Math., 122, 234-236 (1996) · Zbl 0861.05007
[4] Bernard, M.; Straigh, J., Pythagorean triples modulo a prime, Pi Mu Epsilon J., 13, 10, 651-659 (2014) · Zbl 1360.11058
[5] Brualdi, R. A.; Friedland, S.; Pothen, A., The sparse basis problem and multilinear algebra, SIAM J. Matrix Anal. Appl., 16, 1, 1-20 (1995) · Zbl 0824.15003
[6] Davvaz, B.; Leoreanu-Fotea, V., \(n\)-hypergroups and binary relations, European J. Combin., 29, 5, 1207-1218 (2008) · Zbl 1179.20070
[7] Dörnte, W., Untersuchungen über einen verallgemeinerten gruppen begriff, Math. Z., 29, 1, 1-19 (1929) · JFM 54.0152.01
[8] Dudek, W. A., Remarks to Glazek’s results on \(n\)-ary groups, Discuss. Math. Gen. Algebra Appl., 27, 4861-4876 (2007) · Zbl 1153.20050
[9] Eliahou, S.; Lecouvey, C., Matching in arbitrary groups, Adv. in Appl. Math., 40, 219-224 (2008) · Zbl 1160.20304
[10] Eliahou, S.; Lecouvey, C., Matching subspaces in a field extension, preprint · Zbl 1156.20306
[11] Fan, C. K.; Losonczy, J., Matching and canonical forms for symmetric tensors, Adv. Math., 117, 228-238 (1996) · Zbl 0847.05079
[12] Friedland, S., Matrices: Algebra, Analysis and Applications (2015), World Scientific
[13] Friedland, S.; Aliabadi, M., Linear Algebra and Matrices (2018), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 1451.15001
[14] Friedland, S.; Li, Q.; Schonfeld, D., Compressive sensing of sparse tensors, IEEE Trans. Image Process., 23, 10, 4438-4447 (2014) · Zbl 1374.94514
[15] Hall, P., On representatives of subsets, J. Lond. Math. Soc., 10, 26-30 (1935) · JFM 61.0067.01
[16] Lenstra, H. W.; Schoof, J., Primitive normal basis over finite fields, Math. Comp., 48, 217-231 (1987) · Zbl 0615.12023
[17] Losonczy, J., On matching in groups, Adv. in Appl. Math., 20, 385-391 (1998) · Zbl 0944.20039
[18] Malik, D. S.; Mordeson, J. N.; Sen, M. K., Fundamentals of Abstract Algebra (1999), Mc Graw Hill
[19] Rado, R., A theorem on independence relation, Q. J. Math. Oxford Ser., 13, 83-89 (1942) · Zbl 0063.06369
[20] Wakeford, E. K., On canonical forms, Proc. Lond. Math. Soc., 18, 403-410 (1918-1919) · JFM 47.0880.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.