On Groups of Order p&lt;sup&gt;2&lt;/sup&gt;

Sinefakopoulos, Achilleas

doi:10.2307/2691266

Cited by 2 publications

(2 citation statements)

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“…For instance, it implies that any finite field must have prime power order [33, Theorem 6.12]. Certain classification theorems of finite groups arise as an application of Lagrange's theorem [25,26,43,57]. Further, Fermat's little theorem and Euler's theorem may be viewed as a consequence of this theorem.…”

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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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

View full text Add to dashboard Cite

show abstract

“…For instance, it is used to prove that any finite field must have prime power order. Certain classification theorems of finite groups arise as an application of Lagrange's theorem [9,10,17]. Further, Fermat little's theorem and Euler's theorem may be viewed as a consequence of this theorem.…”

Section: Introductionmentioning

confidence: 99%