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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2010
    • (edited Jun 22nd 2010)

    reformatted the entry group a little, expanded the Examples-section a little and then pasted in the group-related “counterexamples” from counterexamples in algebra. Mainly to indicate how I think this latter entry should eventually be used to improve the entries that it refers to.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeJun 23rd 2010

    I removed the counterexample which was not about group theory (and clarified the header in counterexamples in algebra).

    • CommentRowNumber3.
    • CommentAuthorjesuslop
    • CommentTimeDec 21st 2019

    Minor correction, about “The reason is that two functors…” to change that the η h\eta_h natural transformation is between the delooped version of the parallel group homomorphisms, instead of the homomorphisms themselves.

    diff, v62, current

    • CommentRowNumber4.
    • CommentAuthorbezem
    • CommentTimeJan 27th 2020
    Why is the inverse operation not part of the structure of a group (instead of existence of inverses as a property of a monoid)?
    For a topological group it is, and the inverse operation is required to be continuous.
    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 27th 2020

    Yes, the definition section is arguably a little lacking there, even though the article redeems itself later, down in Internalization. So I’ve fixed it along these lines.

    diff, v63, current

  1. linking to pregroup grammar


    diff, v64, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 17th 2021


    The original article that gives a definition equivalent to the modern definition of a group is

    • Heinrich Weber, Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist. Mathematische Annalen 20:3 (1882), 301–329. doi.

    diff, v65, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2022

    added pointer to

    diff, v70, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2022
    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2023

    added pointer to:

    diff, v72, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2023

    and this pointer:

    diff, v72, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2023
    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2023

    added pointer to:

    diff, v74, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2023

    added pointer to:

    diff, v75, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeFeb 5th 2023
    • (edited Feb 5th 2023)

    added historical pointer to:

    (That’s the letter that is commonly cited as the historical origin of the term “group”. But re-reading this now, it seems that this letter does not dwell on saying what a “group” is meant to be, but speaks as if author and recipient both already know of and agree on this notion?)

    diff, v77, current

  2. Add a straightforward definition of a group. The monoidal definition is opaque (despite being useful) and does not exhibit the properties of a group well.


    diff, v81, current

  3. I made a mistake in my initial definition: the existence of a left identity + a right inverse is not sufficient to define a group. This definition now tracks with the standard one.


    diff, v83, current

    • CommentRowNumber18.
    • CommentAuthorʇɐ
    • CommentTimeApr 25th 2024
    • (edited Apr 25th 2024)

    A left identity and left inverses w.r.t. it are, though, if you’re willing to bring in idempotents and other monoid terminology. :) (I agree that the two-sided definition is better, cf. An example of a semigroup with left identities and right inverses is the operation xy=yx \circ y = y, BTW.)