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. https://math.stackexchange.com/questions/65239/right-identity-and-right-inverse-in-a-semigroup-imply-it-is-a-group#comment154118_65261. An example of a semigroup with left identities and right inverses is the operation $x \circ y = y$, BTW.)

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.

JJ

]]>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.

JJ

]]>added historical pointer to:

- Évariste Galois,
*letter to Auguste Chevallier*, (May 1832)

(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?)

]]>added pointer to:

- Martín Escardó,
*Groups*, §33.10 in:*Introduction to Univalent Foundations of Mathematics with Agda*[arXiv:1911.00580, webpage]

added pointer to:

]]>and these two:

]]>and this pointer:

- Lean Community –> mathlib –> algebra.group.defs –> group

added pointer to:

- Farida Kachapova,
*Formalizing groups in type theory*[arXiv:2102.09125]

added pointer to:

- Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Chapter 4 of:
*Symmetry*(2021) $[$pdf$]$

added pointer to

- bananaspace:
*群*(Chinese)

Added:

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.

linking to pregroup grammar

DavidWhitten

]]>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.

]]>For a topological group it is, and the inverse operation is required to be continuous. ]]>

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

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

]]>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.

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