On loops with cyclic inner mapping groups

Kepka, Tomáš; Niemenmaa, Markku

doi:10.1007/bf01198806

Cited by 18 publications

(18 citation statements)

References 4 publications

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“…When we combine Theorem 3.1 with Theorem 2.1 we immediately have the following result in loop theory. [7] On finite loops 483…”

Section: O O P Theoretical Resultsmentioning

confidence: 99%

“…It is not very difficult to see that I(Q) = 1 if and only if Q is an Abelian group. We also know [7,10] that I(Q) is cyclic if and only if Q is an Abelian group. In [9] we showed that for a finite loop Q, I(Q) can not be isomorphic to C n x D, where C n is a cyclic group of order n and I?…”

Section: The Mappings L a (X) -Ax (Left Translation) And R A (X) -Xa mentioning

confidence: 99%

“…In this paper, which is a continuation of [6,7,9,10], we are interested in the structure of inner mapping groups of loops. In [11] we managed to prove that if I{Q) is a finite Abelian group, then M(Q) is solvable.…”

Section: The Mappings L a (X) -Ax (Left Translation) And R A (X) -Xa mentioning

confidence: 99%

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On finite loops whose inner mapping groups are Abelian

Niemenmaa¹

2002

Bull. Austral. Math. Soc.

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ON FINITE LOOPS WHOSE INNER MAPPING GROUPS ARE ABELIANMARKKU NIEMENMAA Loops are nonassociative algebras which can be investigated by using their multiplication groups and inner mapping groups. If the inner mapping group of a loop is finite and Abelian, then the multiplication group is a solvable group. It is clear that not all finite Abelian groups can occur as inner mapping groups of loops. In this paper we show that certain finite Abelian groups with a special structure are not isomorphic to inner mapping groups of finite loops. We use our results and show how to construct solvable groups which are not isomorphic to multiplication groups of loops.

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“…When we combine Theorem 3.1 with Theorem 2.1 we immediately have the following result in loop theory. [7] On finite loops 483…”

Section: O O P Theoretical Resultsmentioning

confidence: 99%

Section: The Mappings L a (X) -Ax (Left Translation) And R A (X) -Xa mentioning

confidence: 99%

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On finite loops whose inner mapping groups are Abelian

Niemenmaa¹

2002

Bull. Austral. Math. Soc.

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“…This resolves the group orders, and it remains to prove that Inn Q cannot be cyclic. Now, nontrivial Inn Q is never cyclic, for every loop Q, by [9] (see also [12] and [6]). We shall observe that this can also be seen directly, without resorting to the general theorem.…”

Section: Proposition 58 Let Q Be a Conjugacy Closed Loop And Letmentioning

confidence: 99%

Structural interactions of conjugacy closed loops

Drápal

2007

Trans. Amer. Math. Soc.

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Abstract. We study conjugacy closed loops by means of their multiplication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication groups, respectively. Put M = {a ∈ Q; L a ∈ R}. We prove that the cosets of A agree with orbits of [L, R], that Q/M ∼ = (Inn Q)/L 1 and that one can define an abelian group on Q/N × Mlt 1 . We also explain why the study of finite conjugacy closed loops can be restricted to the case of N/A nilpotent. Group [L, R] is shown to be a subgroup of a power of A (which is abelian), and we prove that Q/N can be embedded into Aut ([L, R] This paper can be regarded as a sequel to [4] since its main concern rests in exploring how various features of conjugacy closed loops are manifested in their multiplication groups. We shall use several results of [11] and end by setting down the structure of conjugacy closed loops of order pq. Their study was initiated by Goodaire and Robinson in [7] and resumed by Kunen [10], who proved that if a nonassociative conjugacy closed loop of such an order exists, then q < p has to divide p − 1. Kunen also fully described the case q = 2. Here we shall prove that whenever q divides p−1, then there can be constructed, up to isomorphism, exactly one nonassociative conjugacy closed loop of order pq. Its operation can be given by the formulawhere i and j are integers modulo p, r and s are integers modulo q, and γ is a fixed integer whose multiplicative order modulo p is equal to q. Unlike in [10], we shall not construct the loop by considering equalities of loop terms, but we shall derive it from knowledge of the loop's multiplication group. To get the structure of the multiplication group we shall use various facts, some of which have appeared in [11] and [4], and some of which will be proved in this paper. In fact, we shall obtain quite a few structural results, and not all of them will be needed for the case pq. Before turning to their short overview, we shall list those properties of conjugacy closed loops that will be used throughout this paper

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“…If H is a subgroup of a group G then H G denotes the core of H in G (the largest normal subgroup of G contained in H ). Basic facts about connected transversals, loops and their multiplication groups can be found in [3,5,7]. In this paper we consider finite loops and groups only.…”

Section: Introductionmentioning

confidence: 99%

On Connected Transversals to Subgroups Whose Order is a Product of Two Primes

Niemenmaa

1997

European Journal of Combinatorics

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We consider finite groups which have connected transversals to subgroups whose order is a product of two primes p and q. We investigate those values of p and q for which the group is soluble. We can show that the solubility of the group follows if q = 2 and p ≤ 61, q = 3 and p ≤ 31, q = 5 and p ≤ 11. We then apply our results on loop theory and we show that if the inner mapping group of a finite loop has order pq where p and q are as above then the loop is soluble.

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On loops with cyclic inner mapping groups

Cited by 18 publications

References 4 publications

On finite loops whose inner mapping groups are Abelian

On finite loops whose inner mapping groups are Abelian

Structural interactions of conjugacy closed loops

On Connected Transversals to Subgroups Whose Order is a Product of Two Primes

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