A subgroup has equally many left and right cosets

[microreview]

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We prove that a subgroup has the same number of left and right cosets.

https://doi.org/10.31219/osf.io/t4snh

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 137, 2020 [FH]

On the arithmetic of automated theorem proving

[white paper]

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We propose a model to assign prime numbers to axioms and theorems, then by comparing equivalent numbers, it results in new equivalent theorems.

https://doi.org/10.31219/osf.io/wbf85

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 132, 2020 [FC]

A semigroup is a rectangular band if and only if it is nowhere commutative

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/98khs

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 120, 2020 [EQ]

An element of a finite monoid is right invertible if and only if it is left invertible

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/7vm92

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 118, 2020 [EO]

A finite cancellative semigroup is a group

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/34vbp

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 117, 2020 [EN]

A right zero semigroup is left-cancellative

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/g6snd

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 116, 2020 [EM]

A semigroup with a right zero equal to a right identity is trivial

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/zpwb5

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 115, 2020 [EL]

A simple group of order 168 is isomorphic to PGL(2,7)

[question]

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We present a sketch on a problem related to the isomorphism between the simple group of order 168 and the projective general linear group.

https://doi.org/10.31219/osf.io/ydex4

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 114, 2020 [EK]

A simple group of order 60 is isomorphic to a subgroup of S6

[microreview]

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We show that the permutation of six Sylow 5-subgroups by conjugation is a faithful action, so that G is isomorphic to a subgroup of S6.

https://doi.org/10.31219/osf.io/g43yj

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 113, 2020 [EJ]

3-cycle commutator

[microreview]

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We show within the maximum number of steps that if only one point is moved by two permutations, the commutator is a 3-cycle.

https://doi.org/10.31219/osf.io/kxnt8

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 106, 2020 [EC]

Complex complex numbers

[original idea]

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We apply in the complex numbers the same line of thought that led to the very creation of the complex themselves. In addition, we consider multiple imaginary numbers and generalize both ideas altogether.

https://doi.org/10.31219/osf.io/485kj

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 105, 2020 [EB]

Open Mathematics Knowledge Base

[knowledge base]

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We present a list of mathematical results for future implementation in a digital Open Mathematics Knowledge Base.

https://doi.org/10.31219/osf.io/evq8a

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 103, 2020 [DZ]

Mathematical tags of group theory

[microreview]

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I present a collection of mathematical results regarding group theory organized by tags.

https://doi.org/10.31219/osf.io/9xk68

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 101, 2020 [DX]

Counting with groups: a concise approach

[microreview]

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Some definitions and results for counting with groups are presented within the fewest words and symbols as possible.

https://doi.org/10.31219/osf.io/9hdwj

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 100, 2020 [DW]

A list of notations and mathematical definitions for Abstract Algebra organized by tags

[microreview]

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The purpose of this paper is organizational for it presents a list of notations and mathematical definitions used in some results of abstract algebra.

https://doi.org/10.31219/osf.io/u8yx2

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 98, 2020 [DU]

Group actions: a concise approach

[microreview]

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Some definitions and results of group actions are presented within the fewest words and symbols as possible.

https://doi.org/10.31219/osf.io/9dt34

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 97, 2020 [DT]

Homomorphisms: a concise approach

[microreview]

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Some definitions and results of homomorphisms are presented within the fewest words and symbols as possible.

https://doi.org/10.31219/osf.io/7pgcs

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 95, 2020 [DR]

Groups: a concise approach

[microreview]

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Some definitions and results of groups are presented within the fewest words and symbols as possible.

https://doi.org/10.31219/osf.io/bfyhm

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 93, 2020 [DP]

Does zero divide zero?

[original idea]

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We address a discussion whether zero divides zero.

https://doi.org/10.31219/osf.io/5txzg

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 69, 2020 [CQ]

Pre-Requisites for Frobenius Theorem (in real division algebras)

[pre-requisites]

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In order to fully understand the proof of Frobenius theorem (in real division algebras), one must comprehend the following pre-requisites listed in this microarticle.

https://doi.org/10.31219/osf.io/gv8js

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 1, Article 23, 2019 [AW]