Background information:

I recently asked a question about the history of the concepts of the direct sum and the tensor product in group and module theory, and was given a very concise and thorough answer regarding when both the terms and their uses appear in that particular sense for the first time by the user Conifold. Specifically, van der Waerden in 1930-1 appears to be the first one to use the term direct sum as we would use it (though he doesn't use the notation $\oplus$), Murray and Von Neumann are the first one to use the notation $\otimes$ for the tensor product in 1936 (though they do not call it that, and they only concern themselves with certain vector spaces, and that in a roundabout way), and Whitney in 1938 is the one to properly extend it to abelian groups in general and come up with the term tensor product, which he even proves an explanation for:

The most important example of a true tensor product (and the example from which we chose the word "tensor") is probably the following. If $G$ is the tangent vector space at a point of a differentiable manifold, then $G \circ G$ is the space of contravariant tensors of order 2 at the point.

Whitney however uses the notation $\circ$ for the tensor product, but he uses $\oplus$ for the direct sum. Bourbaki later on in the 1940s appears to be the source that definitively puts into place the terminologies direct sum and tensor product and associates with them the notations $\oplus$ and $\otimes$ respectively.

However! Bourbaki still calls what we would call the direct product of two groups for just the product of two groups, and so, I've been trying to find what should be the first use of the term direct product to mean what we would have it mean. I think that I've found a very good candidate, but I need some help with confirming it. Which brings me to my request...

My actual request:

I need the help of someone who can get their hands on specifically the 1st edition of Garrett Birkhoff and Saunders Mac Lane's A Survey of Modern Algebra from 1941. I want to know how they define the direct product in that book.

If they use the modern definition of the direct product in that book, then it should appear in the chapter on Group Theory, in the section on Homomorphisms, and should be on the very first page (or close to) of that section. In the 3rd edition, that is Ch. 6, Sec. 11.

  • $\begingroup$ You sure you have the third edition? Because the third edition on Archive.com has it on p. 138. archive.org/details/surveyofmodernal00birk/mode/2up $\endgroup$ Nov 24, 2021 at 13:28
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    $\begingroup$ Ah, I think I see what the problem is, you have the third edition of Birkhoff's and Mac Lane's Algebra, because there it is indeed defined (for modules) on p. 178. $\endgroup$ Nov 24, 2021 at 13:30
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    $\begingroup$ Whitney defined the tensor product of abelian groups in general, not groups in general. There is no tensor product of general groups. A noncommutative group is not a $\mathbf Z$-module. $\endgroup$
    – KCd
    Nov 24, 2021 at 14:15
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    $\begingroup$ I've found Birkhoff's Lattice Theory (3rd ed 1967, 1st ed 1940), page 8: "The direct product $PQ$ of two posets $P$ and $Q$ is the set of all couples $(x,y)$ such that..." $\endgroup$ Nov 24, 2021 at 15:48
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    $\begingroup$ You might get something from various searches for reviews at JSTOR, such as this search (JSTOR search for "direct product" AND group, between 1920 and 1960, ITEM TYPE = Reviews). Also, try searching for reviews about specific algebra books that you think might be relevant. Keep in mind that some reviews can be extensive and also include information about other books. $\endgroup$ Nov 25, 2021 at 21:32

1 Answer 1


Well, my friend in Sheffield got a hold of a copy of the first edition over at the university library there, and, much to my annoyance, the book mentions neither direct products nor direct unions. Thus, I cannot track back the origin of the term direct product to mean what we currently have it mean to 1941 and Mac Lane and Birkhoff.

Seems like the search goes on...


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