zbMATH — the first resource for mathematics

On multiplication groups of loops. (English) Zbl 0706.20046
From the introduction: While studying the multiplication group of a loop \(Q\) a central role is played by the stabilizer of the neutral element. This subgroup \(I(Q)\) of the multiplication group is called the inner mapping group of \(Q\). It is known that a loop \(Q\) is an Abelian group if and only if \(I(Q)=1\). In this paper some properties of the inner mapping group are described. There is also given a partial answer to the question: What are the multiplication groups of loops? This is closely connected to certain transversal conditions. The multiplication groups of loops are characterized with the aid of these conditions. One of the main results is the theorem: If \(Q\) is a finite loop whose inner mapping group is cyclic then \(Q\) is an Abelian group. Finally it is shown that certain groups are not multiplication groups of loops.
Reviewer: M.Csikós

20N05 Loops, quasigroups
Full Text: DOI
[1] Albert, A.A, Quasigroups I, Trans. amer. math. soc., 54, 507-519, (1943) · Zbl 0063.00039
[2] Albert, A.A, Quasigroups II, Trans. amer. math. soc., 55, 401-419, (1944) · Zbl 0063.00042
[3] Baer, R, Nets and groups, Trans. amer. math. soc., 46, 110-141, (1939) · JFM 65.0819.02
[4] Belousov, V.D, Osnovy teorii kvazigrupp i lup, (1967), Nauka Moskva
[5] Blackburn, N, Finite groups in which the nonnormal subgroups have nontrivial intersection, J. algebra, 3, 30-37, (1966) · Zbl 0141.02401
[6] Bruck, R.H, Contributions to the theory of loops, Trans. amer. math. soc., 60, 245-354, (1946) · Zbl 0061.02201
[7] Bruck, R.H, A survey of binary systems, (1971), Springer-Verlag Berlin/Heidelberg/New York, (third printing) · Zbl 0206.30301
[8] Bruck, R.H; Paige, L, Loops whose inner mappings are automorphisms, Ann. math., 63, 308-323, (1956) · Zbl 0074.01701
[9] Denes, J; Keedwell, A.D, Latin squares and their applications, (1974), Akademiai Kiado Budapest · Zbl 0283.05014
[10] {\scA. Drapal}, Private communication.
[11] Drapal, A; Kepka, T, Alternating groups and Latin squares, European J. combin., 10, 175-180, (1989) · Zbl 0673.20038
[12] Heineken, H; Mohamed, I.J, A group with trivial centre satisfying the normalizer condition, J. algebra, 10, 368-376, (1968) · Zbl 0167.29001
[13] Huppert, B, Endlich gruppen I, (1967), Springer-Verlag Berlin/Heidelberg/New York
[14] Ihringer, T, On multiplication groups of quasigroups, European J. combin., 5, 137-141, (1984) · Zbl 0537.20043
[15] Ihringer, T, Quasigroups, loops and centralizer rings, contr. to general algebra 3, ()
[16] Johnson, K.W, Transversals, S-rings and centralizer rings of groups, () · Zbl 0463.20004
[17] Kepka, T, Multiplication groups of some quasigroups, (), 459-465
[18] T. Kepka, Quasigroups having at most three inner mappings, Acta Univ. Carolin.—Math. Phys., to appear. · Zbl 0702.20053
[19] Kepka, T; Niemenmaa, M, On conjugacy classes in finite loops, Bull. austral. math. soc., 38, 171-176, (1988) · Zbl 0644.20039
[20] Smith, J.D, Multiplication groups of quasigroups, () · Zbl 0335.20035
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.