On multiplication groups of left conjugacy closed loops

Aleš Drápal

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 2, page 223-236
  • ISSN: 0010-2628

Abstract

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A loop Q is said to be left conjugacy closed (LCC) if the set { L x ; x ∈ Q } is closed under conjugation. Let Q be such a loop, let ℒ and ℛ be the left and right multiplication groups of Q , respectively, and let Inn Q be its inner mapping group. Then there exists a homomorphism ℒ → Inn Q determined by L x ↦ R x - 1 L x , and the orbits of [ ℒ , ℛ ] coincide with the cosets of A ( Q ) , the associator subloop of Q . All LCC loops of prime order are abelian groups.

How to cite

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Drápal, Aleš. "On multiplication groups of left conjugacy closed loops." Commentationes Mathematicae Universitatis Carolinae 45.2 (2004): 223-236. <http://eudml.org/doc/249375>.

@article{Drápal2004,
abstract = {A loop $Q$ is said to be left conjugacy closed (LCC) if the set $\lbrace L_x; x \in Q\rbrace $ is closed under conjugation. Let $Q$ be such a loop, let $\mathcal \{L\}$ and $\mathcal \{R\}$ be the left and right multiplication groups of $Q$, respectively, and let $\operatorname\{Inn\} Q$ be its inner mapping group. Then there exists a homomorphism $\mathcal \{L\} \rightarrow \operatorname\{Inn\} Q$ determined by $L_x \mapsto R^\{-1\}_xL_x$, and the orbits of $[\mathcal \{L\}, \mathcal \{R\}]$ coincide with the cosets of $A(Q)$, the associator subloop of $Q$. All LCC loops of prime order are abelian groups.},
author = {Drápal, Aleš},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {left conjugacy closed loop; multiplication group; nucleus; left conjugacy closed loops; multiplication groups; nuclei; inner mapping groups},
language = {eng},
number = {2},
pages = {223-236},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On multiplication groups of left conjugacy closed loops},
url = {http://eudml.org/doc/249375},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Drápal, Aleš
TI - On multiplication groups of left conjugacy closed loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 2
SP - 223
EP - 236
AB - A loop $Q$ is said to be left conjugacy closed (LCC) if the set $\lbrace L_x; x \in Q\rbrace $ is closed under conjugation. Let $Q$ be such a loop, let $\mathcal {L}$ and $\mathcal {R}$ be the left and right multiplication groups of $Q$, respectively, and let $\operatorname{Inn} Q$ be its inner mapping group. Then there exists a homomorphism $\mathcal {L} \rightarrow \operatorname{Inn} Q$ determined by $L_x \mapsto R^{-1}_xL_x$, and the orbits of $[\mathcal {L}, \mathcal {R}]$ coincide with the cosets of $A(Q)$, the associator subloop of $Q$. All LCC loops of prime order are abelian groups.
LA - eng
KW - left conjugacy closed loop; multiplication group; nucleus; left conjugacy closed loops; multiplication groups; nuclei; inner mapping groups
UR - http://eudml.org/doc/249375
ER -

References

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  1. Basarab A.S., Klass LK-lup, Matematicheskie issledovanija 120 (1991), 3-7. (1991) MR1121425
  2. Basarab A.S., IK-loops, Quasigroups and related systems 4 (1997), 1-7. (1997) Zbl0945.20037MR1767513
  3. Belousov V.D., Osnovy teorii kvazigrupp i lup, Nauka, Moskva, 1967. MR0218483
  4. Bruck R.H., A Survey of Binary Systems, Springer-Verlag, 1971. Zbl0141.01401MR0093552
  5. Drápal A., Conjugacy closed loops and their multiplication groups, J. Algebra 272 (2004), 838-850. (2004) Zbl1047.20049MR2028083
  6. Drápal A., Structural interactions of conjugacy closed loops, submitted. 
  7. Goodaire E.G., Robinson D.A., A class of loops which are isomorphic to all loop isotopes, Canad. J. Math. 34 (1982), 662-672. (1982) Zbl0467.20052MR0663308
  8. Kiechle H., Nagy G.P., On the extension of involutorial Bol loops, Abh. Math. Sem. Univ. Hamburg 72 (2002), 235-250. (2002) Zbl1016.20051MR1941556
  9. Kinyon M.K., Kunen K., Phillips J.D., Diassociativity in conjugacy closed loops, Comm. Algebra 32 (2004), 767-786. (2004) Zbl1077.20076MR2101839
  10. Kunen K., The structure of conjugacy closed loops, Trans. Amer. Math. Soc. 352 (2000), 2889-2911. (2000) Zbl0962.20048MR1615991
  11. Nagy P., Strambach K., Loops as invariant sections in groups, and their geometry, Canad. J. Math. 46 (1994), 1027-1056. (1994) Zbl0814.20055MR1295130
  12. Matievics I., Geometries over universal left conjugacy closed quasifields, Geom. Dedicata 65 (1997), 127-133. (1997) Zbl0881.51005MR1451967
  13. Soikis L.R., O specialnych lupach, in Voprosy teorii kvazigrupp i lup (V.D. Belousov, ed.), Akademia Nauk Moldav. SSR, Kishinev, 1970, pp. 122-131. MR0274626

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