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This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in analysis and Counterexamples in topology, so I think it's time for: Counterexamples in algebra.

Now, algebra is quite broad, and I'm new at this, so if I need to narrow this then I will- just let me know. At the moment I'm looking for counterexamples in all areas of algebra: finite groups, representation theory, homological algebra, Galois theory, Lie groups and Lie algebras, etc. This might be too much, so a moderator can change that.

These counterexamples can illuminate a definition (e.g. a projective module that is not free), illustrate the importance of a condition in a theorem (e.g. non-locally compact group that does not admit a Haar measure), or provide a useful counterexample for a variety of possible conjectures (I don't have an algebraic example, but something analogous to the Cantor set in analysis). I look forward to your responses!

You can also add your counter-examples to this nLab page: http://ncatlab.org/nlab/show/counterexamples+in+algebra

(the link to that page is currently "below the fold" in the comment list so I (Andrew Stacey) have added it to the main question)

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    $\begingroup$ My feeling is that this question is far too broad. $\endgroup$
    – Andy Putman
    Commented Jun 21, 2010 at 23:11
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    $\begingroup$ I like that the question is general. I think if it's narrowed too much we won't get as many interesting responses. All of the big list type questions that have been successful have been fairly general, so I don't think it hurts as long as we aren't swarmed with questions like this. $\endgroup$
    – jeremy
    Commented Jun 22, 2010 at 0:23
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    $\begingroup$ Meta discussion: tea.mathoverflow.net/discussion/459/counterexamples-in-algebra $\endgroup$
    – Andrew Stacey
    Commented Jun 22, 2010 at 7:54
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    $\begingroup$ Whilst I like lists of counterexamples, I don't think that MO is an appropriate place for one. I've explained why in the meta discussion (NB: please vote for the comment linking to the meta discussion so that it appears "above the fold"). I think that this would work so much better as a wiki page. So I've started one on the nLab: ncatlab.org/nlab/show/counterexamples+in+algebra Obviously, as I'm not an algebraist I didn't understand everything and have probably left out a lot of information in copying it over. I recommend closing this question and redirecting to that nLab page. $\endgroup$
    – Andrew Stacey
    Commented Jun 22, 2010 at 8:33
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    $\begingroup$ Andrew, why not keep the question open, in order to generate the examples here that can then be more sensibly organized on your page? It seems likely to me that you will get a lot of good examples with this question that might otherwise be missed. $\endgroup$
    – Joel David Hamkins
    Commented Jun 22, 2010 at 13:16

63 Answers 63

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OP: [...] counterexamples can illuminate a definition (e.g. a projective module that is not free), [...]

Indeed, let our ring $\ \mathcal R\ $ be the the ring of all continuous functions from the Euclidean sphere $\ \mathbb S\ :=\ \mathbb S^2\ $ (or more generally, $\ \mathbb S\ :=\ \mathbb S^{2\cdot n},\ $ where $\ n\in\mathbb N).\ $ Then module $\ \mathbb T\ $ of all continuous vector fields that are tangent to $\ \mathbb S\ $ is a direct summand of free module $\ \mathcal R^3\ $ hence $\ \mathbb T\ $ is projective but it is not free.

The last property of $\ \mathbb T\ $ that states that $\ \mathbb T\ $ is not free is implied by the Karol Borsuk's theorem about the unruly hair on sphere $\ \mathbb S\ $ that is impossible to brush smoothly.

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This one is only about terminology, but while the topic is couterexamples in Algebra so it's tempting to give this one: Lie algebra is not an algebra.

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    $\begingroup$ Sure it is. It is just not an associative algebra. $\endgroup$
    – Tobias Kildetoft
    Commented Dec 15, 2014 at 10:59
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    $\begingroup$ It's an algebra and it's not an algebra. Mathematicians seem to find practical to use "algebra" in several distinct meanings, which are more or less obvious to guess according to the context, when they are not specified. In practice, it is very common that people deal with together both associative unital algebras and Lie algebras, and call the first simply "algebras" and the second "Lie algebras" (e.g., search stuff about universal enveloping algebras, since it relates the two). $\endgroup$
    – YCor
    Commented Oct 31, 2016 at 4:51
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Videque

  1. Counterexamples in X
  2. Counterexamples in Clifford Algebras
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