A Note on Simple Groups and Simple Loops

Phillips, J. D.; Baik, Young Gheel; Johnson, Johnson David L.; Kim, Ann Chi

doi:10.1515/9783110807493-022

Cited by 5 publications

(4 citation statements)

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“…For a general overview about the relation between loops and groups we refer to [2,10,13,14,15]. The reader interested in the history of loop theory should have a look at the article by Pflugfelder [12] and those who wish to know about the applications of nonassociative algebras are advised to read the articles [3,4,8].…”

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

confidence: 99%

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

confidence: 99%

On finite loops whose inner mapping groups are Abelian

Niemenmaa¹

2002

Bull. Austral. Math. Soc.

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“…For basic facts about loop theory and its history the reader is advised to consult the articles by Bruck [1], Pflugfelder [18] and Smith [20]. For more recent results in loop theory we recommend the articles by Kinyon, Kunen and Phillips [11,12,19]. …”

Section: Introductionmentioning

confidence: 99%

Finite loops with dihedral inner mapping groups are solvable

Niemenmaa

2004

Journal of Algebra

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“…Our notation is standard. Basic facts about the connected transversals loops and then multiplication groups can be found in [2], [9], [16], [17], [18]. Following Kepka (see [4]) we define the next two conditions (a) and (b) for an abelian group H .…”

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confidence: 99%

On connected transversals to abelian subgroups and loop theoretical consequences

Csörgő

2006

Arch. Math.

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In this paper one of our questions is the following: Which finite abelian groups are (are not) isomorphic to inner mapping groups of loops? It is well known that if the inner mapping group of a finite loop Q is abelian, then Q is centrally nilpotent. The other question is: Which properties of abelian inner mapping groups imply the central nilpotency of class at most two of the loop? After reminding the reader of the known results we show new ones. To solve these problems we transform them into group theoretical problems, then using connected transversals we get some answer. Introduction. Q is a loop if it isa quasigroup with neutral element. The functions L a (x) = ax (left translation) and R a (x) = xa (right translation) are permutations on the elements of Q for every a ∈ Q. The permutation group generated by left and right translations M(Q) = L a , R a /a ∈ Q is called the multiplication group of Q. Denote by I (Q) the stabilizer of the neutral element in M(Q). I (Q) is a subgroup of M(Q) and it is called the inner mapping group of Q. It is clear that M(Q) is transitive and the stabilizers of the elements of Q are conjugate in M(Q). Bruck [2] introduced this connection between loops and groups, and he investigated the structure of loops by using group theory.There are several problems concerning the loops with abelian inner mapping group. One of them is the following: a) Which finite abelian groups can (cannot) occur as the inner mapping group of loops? The question which finite abelian groups are possible as inner automorphism groups of groups was completely solved by Baer [1]. The result is as follows:Let G be a finite abelian group and let G = C 1 × · · · × C n be the direct product of cyclic groups such that |C i+1 | divides |C i | (i = 1, . . . , n − 1). Then there exists a group H such that Inn(H ) ∼ = G if and only if n 2 and |C 1 | = |C 2 |.

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A Note on Simple Groups and Simple Loops

Cited by 5 publications

References 0 publications

On finite loops whose inner mapping groups are Abelian

On finite loops whose inner mapping groups are Abelian

Finite loops with dihedral inner mapping groups are solvable

On connected transversals to abelian subgroups and loop theoretical consequences

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