On Twisted Subgroups and Bol Loops of Odd Order

Foguel, Tuval; Kinyon, Michael; Phillips, J. D.

doi:10.1216/rmjm/1181069494

Cited by 31 publications

(35 citation statements)

References 28 publications

(68 reference statements)

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“…The next result appears in various places in the literature (e.g., Proposition 5.2 in [11] and Lemma 2.5 in [18]) but we supply a proof for completeness.…”

Section: Bol Loopsmentioning

confidence: 90%

“…Much of the following two lemmas appears in Remark 5.12 in [11], with proofs in a different language appearing in Section 4 of [12]: 6.6. Let X be a Bruck loop and set (X) = (G, H, K).…”

Section: Let ξ = (G H K) Be a Bol Loop Folder Then Each Subfolder mentioning

confidence: 99%

“…[18] and [11]), a loop X with envelope (X) = (G, H, K) is a Bol loop iff K is a twisted subgroup of G. Section 6 contains a proof of this fact for completeness, and Section 5 contains some discussion of twisted subgroups. After an initial reduction, twisted subgroups of groups G can be described in terms of automorphisms τ of G with τ 2 = 1, as subsets of the set K(τ ) = g ∈ G: g τ = g −1 of elements of G inverted by τ .…”

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confidence: 99%

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On Bol loops of exponent 2

Aschbacher

2005

Journal of Algebra

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Communicated by Richard LyonsLet X be a magma; that is X is a set together with a binary operation • on X. For each x ∈ X we obtain maps R(x) and L(x) on X defined by R(x) : y → y • x and L(x) : y → x • y called right and left translation by x, respectively. A loop is a magma X with an identity 1 such that R(x) and L(x) are permutations of X for all x ∈ X. In essence loops are groups without the associative axiom.

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“…The next result appears in various places in the literature (e.g., Proposition 5.2 in [11] and Lemma 2.5 in [18]) but we supply a proof for completeness.…”

Section: Bol Loopsmentioning

confidence: 90%

“…Much of the following two lemmas appears in Remark 5.12 in [11], with proofs in a different language appearing in Section 4 of [12]: 6.6. Let X be a Bruck loop and set (X) = (G, H, K).…”

Section: Let ξ = (G H K) Be a Bol Loop Folder Then Each Subfolder mentioning

confidence: 99%

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confidence: 99%

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On Bol loops of exponent 2

Aschbacher

2005

Journal of Algebra

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“…Foguel et al [23] proved that left Bol loops of odd order satisfy the strong Lagrange property. It is, however, still an open problem whether or not finite Bol loops satisfy the Lagrange property [20, p. 592].…”

Section: The Lagrange Propertymentioning

confidence: 99%

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

Suksumran

2016

Essays in Mathematics and Its Applications

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Using the Clifford algebra formalism, we show that the unit ball of a real inner product space equipped with Einstein addition forms a uniquely 2-divisible gyrocommutative gyrogroup or a B-loop in the loop literature. One notable result is a compact formula for Einstein addition in terms of Möbius addition. In the second part of this paper, we show that the symmetric group of a gyrogroup admits the gyrogroup structure, thus obtaining an analog of Cayley's theorem for gyrogroups. We examine subgyrogroups, gyrogroup homomorphisms, normal subgyrogroups, and quotient gyrogroups and prove the isomorphism theorems. We prove a version of Lagrange's theorem for gyrogroups and use this result to prove that gyrogroups of particular order have the Cauchy property.

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“…We refer to left Bol loops simply as Bol loops for the balance of the paper. In Bol loops, each element has a unique two-sided inverse element, denoted by x −1 , satisfying [6,25] and Bruck loops [1,12,13,14,18]; they are two of the most important and widely investigated classes of loops.…”

mentioning

confidence: 99%