On Bruck loops of 2-power exponent

Baumeister, Barbara; Stroth, Gernot; Stein, Alexander

doi:10.1016/j.jalgebra.2010.10.033

Cited by 3 publications

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“…We could extend our results to Bruck loops mainly because of the following lemma. Statement (4) is not formulated in 2.15 of [BSS10], but it is an easy consequence of (1)-(3). [BSS10,2.15 (2)-(4)]. )…”

Section: Previous Resultsmentioning

confidence: 99%

“…Statement (4) is not formulated in 2.15 of [BSS10], but it is an easy consequence of (1)-(3). [BSS10,2.15 (2)-(4)]. ) Let (G, H, K ) be a Bruck folder.…”

Section: Previous Resultsmentioning

confidence: 99%

“…Now not only the class of Bruck loops is understood better, but also the more general class of Bol-A r -loops (for the definition of an A r -loop see [BSS10] …”

Section: Corollary 12 Let X Be a Finite Bruck Loop With Enveloping mentioning

confidence: 99%

“…The proof of Theorem 1 is based on Theorem 1 of [BSS10], which reduces the classification of finite Bruck loops of exponent 2 to a group theoretic question, on the solution of that question [S09] and on the structure theorem of Bruck loops, Theorem 1 in [AKP06].…”

Section: The Structure Of the Papermentioning

confidence: 99%

“…If (G, H, K ) is a BX2P-folder, then the loop related to it is a Bruck loop of 2-power exponent [BSS10]. Moreover, notice that the subgroup H of the Baer envelope induces automorphisms on X in a Bruck loop.…”

Section: Definitions and Notationmentioning

confidence: 99%

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The finite Bruck loops

Baumeister

Stein

2011

Journal of Algebra

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We continue the work of Aschbacher, Kinyon and Phillips (2006) [AKP06] as well as of Glauberman (1964Glauberman ( , 1968 [G64,G68] by describing the structure of the finite Bruck loops. We show that a finite Bruck loop X is the direct product of a Bruck loop of odd order with either a soluble Bruck loop of 2-power order or a product of loops related to the groups PSL 2 (q), q = 9 or q 5 a Fermat prime.The latter possibility does occur as is shown in Nagy (2008) [N08] and Baumeister and Stein (2010) [BS10]. As corollaries we obtain versions of Sylow's, Lagrange's and Hall's Theorems for loops.

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Section: Previous Resultsmentioning

confidence: 99%

“…Statement (4) is not formulated in 2.15 of [BSS10], but it is an easy consequence of (1)-(3). [BSS10,2.15 (2)-(4)]. ) Let (G, H, K ) be a Bruck folder.…”

Section: Previous Resultsmentioning

confidence: 99%

“…Now not only the class of Bruck loops is understood better, but also the more general class of Bol-A r -loops (for the definition of an A r -loop see [BSS10] …”

Section: Corollary 12 Let X Be a Finite Bruck Loop With Enveloping mentioning

confidence: 99%

Section: The Structure Of the Papermentioning

confidence: 99%

Section: Definitions and Notationmentioning

confidence: 99%

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The finite Bruck loops

Baumeister

Stein

2011

Journal of Algebra

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Involutive groups, unique 2-divisibility, and related gyrogroup structures

Suksumran

2017

J. Algebra Appl.

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In this paper, we establish a strong connection between groups and gyrogroups, which provides the machinery for studying gyrogroups via group theory. Specifically, we prove that there is a correspondence between the class of gyrogroups and a class of triples with components being groups and twisted subgroups. This in particular provides a construction of a gyrogroup from a group with an automorphism of order two that satisfies the uniquely 2-divisible property. We then present various examples of such groups, including the general linear groups over [Formula: see text] and [Formula: see text], the Clifford group of a Clifford algebra, the Heisenberg group on a module, and the group of units in a unital C[Formula: see text]-algebra. As a consequence, we derive polar decompositions for the groups mentioned previously.

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Pseudoautomorphisms of Bruck loops and their generalizations

Greer,

Kinyon

2012

Preprint

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We show that in a weak commutative inverse property loop, such as a Bruck loop, if α is a right [left] pseudoautomorphism with companion c, then c [c 2 ] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. We also show that every pseudoautomorphism of a commutative inverse property loop is an automorphism, generalizing a well-known result of Bruck.A loop (Q, •) consists of a set Q with a binary operation • : Q × Q → Q such that (i) for all a, b ∈ Q, the equations ax = b and ya = b have unique solutions x, y ∈ Q, and (ii) there exists 1 ∈ Q such that 1x = x1 = x for all x ∈ Q. We denote these unique solutions by x = a\b and y = b/a, respectively. For x ∈ Q, define the right and left translations by x by, respectively, yR x = yx and yL x = xy for all y ∈ Q. That these mappings are permutations of Q is essentially part of the definition of loop. Standard reference in loop theory are [7,13].A triple (α, β, γ) of permutations of a loop Q is an autotopism if for all x, y ∈ Q, xα • yβ = (xy)γ. The set Atp(Q) of all autotopisms of Q is a group under composition. Of particular interest here are the three subgroupsand so every element of Atp λ (Q) has the form (βL a , β, βL a ) for some a ∈ Q. Conversely, it is easy to see that if a triple of permutations of that form is an autotopism, then 1β = 1.By similar arguments for the other two cases, we have the following characterizations:Since these special types of autotopisms are entirely determined by a single permutation and an element of the loop, it is customary to focus on those instead of on the autotopisms themselves. This motivates the following definitions.Let Q be a loop. If β ∈ Sym(Q) and a ∈ Q satisfy a • (xy)β = (a • xβ)(yβ) (1) 2010 Mathematics Subject Classification. 20N05.

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On Bruck loops of 2-power exponent

Cited by 3 publications

References 8 publications

The finite Bruck loops

The finite Bruck loops

Involutive groups, unique 2-divisibility, and related gyrogroup structures

Pseudoautomorphisms of Bruck loops and their generalizations

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