A subgroup has equally many left and right cosets

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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]

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

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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

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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

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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

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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

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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

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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

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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]

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

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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

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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

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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

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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

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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]