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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