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Keywords:
central quasigroups; $T$-quasigroups; multiplication groups; Frobenius groups; quasigroups isotopic to Abelian groups
central quasigroups; $T$-quasigroups; multiplication groups; Frobenius groups; quasigroups isotopic to Abelian groups
Summary:
If $Q$ is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group $\mathop {\mathrm Mlt}Q$ is a Frobenius group. Conversely, if $\mathop {\mathrm Mlt}Q$ is a Frobenius group, $Q$ a quasigroup, then $Q$ has to be isotopic to an Abelian group. If $Q$ is, in addition, finite, then it must be a central quasigroup (a $T$-quasigroup).
References:
[1] A. Albert: Quasigroups I. Trans. Amer. Math. Soc. 54 (1943), 507–519. DOI 10.1090/S0002-9947-1943-0009962-7 | MR 0009962 | Zbl 0063.00039
[2] V. D. Belousov: Balanced identities in quasigroups. Matem. sbornik (N.S.) 70 (1966), 55–97. (Russian) MR 0202898 | Zbl 0199.05203
[3] V. D. Belousov: Osnovy teorii kvazigrupp i lup. Nauka, Moskva, 1967. MR 0218483
[4] G. B. Belyavskaya: $T$-quasigroups and the centre of a quasigroup. Matem. Issled. 111 (1989), 24–43. (Russian) MR 1045383
[5] G. B. Belyavskaya: Abelian quasigroups and $T$-quasigroups. Quasigroups and related systems 1 (1994), 1–7. MR 1327941
[6] R. H. Bruck: A Survey of Binary Systems. Springer-Verlag, 1971. MR 0093552
[7] O. Chein, H. O. Pflugfelder and J. D. H. Smith: Quasigroups and Loops: Theory and Applications. Heldermann, Berlin, 1990. MR 1125806
[8] A. Drápal: Multiplication groups of free loops I. Czechoslovak Math. J. 46 (1996), 121–131. MR 1371694
[9] A. Drápal: Multiplication groups of free loops II. Czechoslovak Math. J. 46 (1996), 201–220. MR 1388610
[10] A. Drápal: Multiplication groups of finite free loops that fix at most two points. J. Algebra 235 (2001), 154–175. DOI 10.1006/jabr.2000.8472 | MR 1807660
[11] A. Drápal: Orbits of inner mapping groups. Monatsh. Math. 134 (2002), 191–206. DOI 10.1007/s605-002-8256-2 | MR 1883500
[12] G. Grätzer: Universal Algebra. Van Nostrand, Princeton, 1968. MR 0248066
[13] J. Ježek and T. Kepka: Quasigroups isotopic to a group. Comment. Math. Univ. Carolin. 16 (1975), 59–76. MR 0367103
[14] J. Ježek: Normal subsets of quasigroups. Comment. Math. Univ. Carolin. 16 (1975), 77–85. MR 0367104
[15] J. Ježek: Univerzální algebra a teorie modelů. SNTL, Praha, 1976. MR 0546057
[16] T. Kepka and P. Němec: $T$-quasigroups I. Acta Univ. Carolin. Math. Phys. 12 (1971), 39–49. MR 0320206
[17] T. Kepka and P. Němec: $T$-quasigroups II. Acta Univ. Carolin. Math. Phys. 12 (1971), 31–49. MR 0654381
[18] J. D. H. Smith: Mal’cev Varieties. Springer, Berlin, 1976. MR 0432511 | Zbl 0344.08002
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