There appears to be anachronistic notation in Robinson's book "A Course in the Theory of Groups (Second Edition)", page 105, Exercise 4.2.8.

NB: This is a question.

The Details:

The exercise reads

(Dieudonné). Let $\mu:A\to B$ be a monomorphism of abelian groups and let $G$ be a torsion-free abelian group. Prove that $\mu_*:A\otimes_{\Bbb Z}G\to B\otimes_{\Bbb Z}G$ is a monomorphism where $(a\otimes g)\mu_*=(a\mu)\otimes g.$ [Hint: Reduce first to the case where $G$ is finitely generated and then to the case $G=\Bbb Z$.]

The Question:

What does Robinson mean by "$A\otimes_{\Bbb Z}G$" here exactly? Is it defined earlier on in the text? (I can't find it anywhere . . . )


I believe it is the tensor product. The book doesn't say anything about tensor products until page 131 outside the exercises (as far as I can tell), and even then, they're not defined. It's strange because Robinson goes to the trouble of defining, say, semigroups, and builds up basic concepts, but then expects the reader to know what a tensor product is.

Granted, the book is a graduate textbook. It doesn't define what a vector space is, either, for example, but refers to one in Exercise 4.2.6.

Please help :)

  • 1
    $\begingroup$ I'm assuming they mean $\mu_*(a \otimes g) = \mu(a) \otimes g$ right? And yes, I do think they're referring to tensor products (which is why it's so important for $G$ to be torsion-free). $\endgroup$ Jul 23, 2021 at 22:47
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    $\begingroup$ @MarkSaving: I've seen places where people write group actions on the right. Plain functions, not so much (except when it's not obvious what is the function and what is the argument...). $\endgroup$
    – tomasz
    Jul 23, 2021 at 23:02
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    $\begingroup$ I believe you are right, it is meant to be the tensor product. Maybe the author is assuming that the reader is already familiar with basic linear/commutative algebra? It's not so odd to teach vector spaces before groups, though it is odder to teach modules before groups. In any event, if it's a graduate textbook, they might assume that the reader is actually familiar with the basics of the subject matter (and still partially recount them, if only to ensure that the important stuff is well understood). $\endgroup$
    – tomasz
    Jul 23, 2021 at 23:04
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    $\begingroup$ @MarkSaving: There are several group theory books that were originally written in the 60s and 70s that use "functions on the right" notation (e.g., Hanna Neumann's Varieties of Groups); also, a lot of ring theorists use functions on the right: e.g., they make morphisms of modules look like associativity ($(rm)f = r(mf)$) and makes left $R$-modules into $R$-$\mathrm{End}_R(M)$ bimodules. It drifted in group theory, and people who work a lot with group actions seem to like the notation. $\endgroup$ Jul 23, 2021 at 23:23
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    $\begingroup$ This is the "tensor product over $\mathbb{Z}$": the universal object that codes bilinear maps from $A\times G$.. I don't have the book in front of me (it's in my office), so I can't check if he's defined it, but I can take a look on Monday. $\endgroup$ Jul 23, 2021 at 23:24

1 Answer 1


The tensor product is defined on p. 235 in the chapter on representations (8.4), despite being used earlier.


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