On extraspecial left conjugacy closed loops

Drápal, Aleš

doi:10.1016/j.jalgebra.2006.04.018

Cited by 11 publications

(5 citation statements)

References 18 publications

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“…This equation already hints at combinatorial polarization (see below). For p odd, symplectic conjugacy closed p-loops were characterized by Drápal [8] by means of modifications of symplectic Abelian p-groups: Theorem 2.7 (Theorem 7.1 of [8]) Let p be an odd prime, and let (G, +) be an Abelian group containing a subgroup F of order p such that V = G/F is an elementary Abelian group. Let f : V 3 → F be a symmetric trilinear form, and let g : V 2 → F be an alternating bilinear form.…”

Section: Conjugacy Closed Loops and Symmetry Of The Associator Mappingmentioning

confidence: 99%

“…Then V θ is isomorphic to V ϑ if and only if (P θ , C θ , A θ ) is conjugate to (P ϑ , C ϑ , A ϑ ) under GL(V ), that is, there is ϕ ∈ GL(V ) such that P θ (u) = P ϑ (ϕu), C θ (u, v) = C ϑ (ϕu, ϕv), and A θ (u, v, w) = A ϑ (ϕu, ϕv, ϕw) for every u, v, w ∈ V . Theorem 9.6 (Theorem 7.2 of [8]) Let V θ 1 , V θ 2 be odd code loops via characteristic trilinear forms f 1 , f 2 : V 3 → F p , respectively. Then there exists an isomorphism With u = λ i e i , v = μ j e j , w = ν k e k , we have…”

Section: Corollary 94 Assume That P > 3 Then An Odd Code Loop Is Pomentioning

confidence: 99%

“…Since p divides (p − 1)p(2p − 1)/12 when p > 3, we get R p x (y) = px + y for p > 3, and we see that However, there exists a symplectic conjugacy closed 3-loop of order 9 in which x(xx) = 1 holds but (xx)x = 1 does not. To see what happens when (10.3) is dropped from Definition 3.3, see [8].…”

Section: The Mapping Pmentioning

confidence: 99%

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Code loops in both parities

Drápal

Vojtěchovský

2009

J Algebr Comb

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Section: Conjugacy Closed Loops and Symmetry Of The Associator Mappingmentioning

confidence: 99%

Section: Corollary 94 Assume That P > 3 Then An Odd Code Loop Is Pomentioning

confidence: 99%

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Code loops in both parities

Drápal

Vojtěchovský

2009

J Algebr Comb

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“…loops. Some studies on them can be found in Drápal [17,18,19,20,21], Csörgő and Drápal [16], Csörgő [15], Kinyon and Kunen [44,45], Kinyon et. al.…”

Section: Introductionmentioning

confidence: 99%

Holomorphy of Osborn loops

Isere¹,

Adeniran²,

Jaiyeola³

2015

Annals of West University of Timisoara - Mathematics and Computer Science

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Let $(L,\cdot)$ be any loop and let $A(L)$ be a group of automorphisms of $(L,\cdot)$ such that $\alpha$ and $\phi$ are elements of $A(L)$. It is shown that, for all $x,y,z\in L$, the $A(L)$-holomorph $(H,\circ)=H(L)$ of $(L,\cdot)$ is an Osborn loop if and only if $x\alpha (yz\cdot x\phi^{-1})= x\alpha (yx^\lambda\cdot x) \cdot zx\phi^{-1}$. Furthermore, it is shown that for all $x\in L$, $H(L)$ is an Osborn loop if and only if $(L,\cdot)$ is an Osborn loop, $(x\alpha\cdot x^{\rho})x=x\alpha$, $x(x^{\lambda}\cdot x\phi^{-1})=x\phi^{-1}$ and every pair of automorphisms in $A(L)$ is nuclear (i.e. $x\alpha\cdot x^{\rho},x^{\lambda}\cdot x\phi\in N(L,\cdot )$). It is shown that if $H(L)$ is an Osborn loop, then $A(L,\cdot)= \mathcal{P}(L,\cdot)\cap\Lambda(L,\cdot)\cap\Phi(L,\cdot)\cap\Psi(L,\cdot)$ and for any $\alpha\in A(L)$, $\alpha= L_{e\pi}=R^{-1}_{e\varrho}$ for some $\pi\in \Phi(L,\cdot)$ and some $\varrho\in \Psi(L,\cdot)$. Some commutative diagrams are deduced by considering isomorphisms among the various groups of regular bijections (whose intersection is $A(L)$) and the nucleus of $(L,\cdot)$.Comment: 17 pages, 12 figure

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“…Some of them can be transferred to the one sided situation of, say, the LCC loops, see [22,9,10]. The recent history of CC loops [8,19,20,11] unwinds from the discovery of Basarab [1] that isotopically invariant LCC loops are abelian groups modulo the left nucleus. Thus Q/N is an abelian group for every CC loop Q, and this is the only difficult property of the conjugacy closed loops among those that are listed above.…”

mentioning

confidence: 99%