Extending the structural homomorphism of LCC loops

Piroska Csörgö

Displaying similar documents to “Extending the structural homomorphism of LCC loops”

Normality, nuclear squares and Osborn identities

Aleš Drápal, Michael Kinyon (2020)

Commentationes Mathematicae Universitatis Carolinae

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Let $Q$ be a loop. If $S \leq Q$ is such that $ϕ (S) \subseteq S$ for each standard generator of Inn $Q$ , then $S$ does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus....

A class of Bol loops with a subgroup of index two

Petr Vojtěchovský (2004)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group and $C_{2}$ the cyclic group of order $2$ . Consider the $8$ multiplicative operations $(x, y) \mapsto {(x^{i} y^{j})}^{k}$ , where $i$ , $j$ , $k \in {- 1, 1}$ . Define a new multiplication on $G \times C_{2}$ by assigning one of the above $8$ multiplications to each quarter $(G \times {i}) \times (G \times {j})$ , for $i, j \in C_{2}$ . We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M (G, 2)$ .

On multiplication groups of left conjugacy closed loops

Aleš Drápal (2004)

Commentationes Mathematicae Universitatis Carolinae

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A loop $Q$ is said to be left conjugacy closed (LCC) if the set ${L_{x}; x \in Q}$ 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 $ℒ \to Inn Q$ determined by $L_{x} \mapsto R_{x}^{- 1} L_{x}$ , 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.

On finite commutative loops which are centrally nilpotent

Emma Leppälä, Markku Niemenmaa (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $Q$ be a finite commutative loop and let the inner mapping group $I (Q) ≅ C_{p^{n}} \times C_{p^{n}}$ , where $p$ is an odd prime number and $n \geq 1$ . We show that $Q$ is centrally nilpotent of class two.