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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.
I removed the counterexample which was not about group theory (and clarified the header in counterexamples in algebra).
Added:
The original article that gives a definition equivalent to the modern definition of a group is
added pointer to:
added pointer to:
and these two:
added pointer to:
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?)
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.)
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