# First appearance of modern definition of a group

What is the first appearance in print of the modern definition of an abstract group? To qualify, it should be a formal definition, contain the word "elements" (so Burnside's 1897 restriction to "operations", where associativity is automatic, does not qualify), and contain the 4 standard properties: closure, associativity, identity, inverses (so Weber's 1882 two-sided cancellation does not qualify).

I find all these requirements in Harold Hilton, An Introduction to the Theory of Groups of Finite Order, published in 1908. Is there an earlier appearance?

• – lhf
Jul 28, 2015 at 0:31
• cf. this Dec 19, 2017 at 22:44

Maybe Dedekind could have done what you are looking for. The sources for the early development apparently include Cauchy, Galois, Abel, Cayley, etc. However, according to Bell "Men of Mathematics" (enjoyable to read, but not a really scholarly work), page 282:

"This abstract point of view is that now current. It was not Cauchy's, but was introduced by Cayley in 1854. Nor were completely satisfactory sets of postulates for groups stated till the first decade of the twentieth century."

Bell says the following on page 518:

"Dedekind was one of the first to appreciate the fundamental importance of the concept of a group in algebra and arithmetic. In this early work Dedekind already exhibited two of the leading characteristics of his later thought, abstractness and generality. Instead of regarding a finite group from the standpoint offered by its representation in terms of substitutions [....], Dedekind defined groups by means of their postulates [....] and sought to derive their properties from this distillation of their essence. This is in the modern manner: abstractness and therefore generality."

I'm not sure if Dedekind is the "first decade of the twentieth century" mathematician Bell was talking about. I think a more interesting secondary question here is when did group theory explicitly include infinite and continuous groups. I'm guessing that in the early days, finite groups were the focus. The combination of the purely algebraic finite groups with analysis to study infinite groups was much later, and I guess Sophus Lie gets most of the credit for that.

From Wikipedia, Walther von Dyck gave the first modern definition of a group.

• Does it satisfy the listed requirements? Jul 25, 2015 at 20:00
• Dyck's definition arguably doesn't satisfy any of the requirements.
– David Callan
Jul 25, 2015 at 21:12
• @DavidCallan oh :/
– mathers101
Jul 25, 2015 at 21:39
• Exactly - Dyck must have defined a Gruppe instead of a group.
– user2255
Jul 27, 2015 at 12:17

"... and the notion of abstract groups was introduced by A.Cayley in three papers starting in 1849..." (from here)

By the way, the original von Dyck's paper 1882 does carry the epigraph of Cayley's quote.

UPDATE: Another historical paper from 2007 "Arthur Cayley, the multiplication table and the notion of abstract group" (in German). Cannot read German myself unfortunately...