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[cross posted from mse]

the class of all finite groups is not closed under produtcs - example: the product over all finite cyclic groups - thus it is not a variety of algebras, ie, it's not axiomatizable by universal sentences.

in fact more is true: it's not axiomatizable at all, by a consequence of the compactness theorem for first order theories, namely, that a theory having models of arbitrarily large finite size - again, take the cyclic groups - must have an infinite model.

proofs of theorems pertaining to finite but not to arbitrary groups have more of an 'arithmetical flavor' than a 'general nonsensical' one, and it's not hard to envision the subject being developed inside first-order number theory.

so what is it that makes the subject of finite groups 'algebraic'? why does it appear on, say, syllabi on undergrad and graduate algebra?

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    $\begingroup$ I suspect you're bowing down to a false god when you treat that kind of definition of "algebra" as if it were something known to be somehow correct. $\endgroup$
    – Michael Hardy
    Commented Apr 12, 2021 at 21:22
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    $\begingroup$ Oh my--when would an undergraduate, or even graduate student, be allowed to read about the classification of finite simple groups? $\endgroup$
    – user52817
    Commented Apr 12, 2021 at 21:27
  • $\begingroup$ not 'correct', of course [definitions don't have truth values], but it seems reasonable to me; it just seems finite group theory is fundamentally 'arithmetical' in character $\endgroup$
    – ac15
    Commented Apr 12, 2021 at 21:28
  • $\begingroup$ @user52817 ideally after understanding 'simplicity' in terms of 'normal subobjects', and these in terms of kernels [either universal-algebraically/model-theoretically or categorially] + understanding simple groups appear naturally in considering composition series $\endgroup$
    – ac15
    Commented Apr 12, 2021 at 21:39
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    $\begingroup$ @ac15 - From where I sit, the entire purpose of general nonsense is that it takes complicated problems, peels away the obfuscation, and exposes their combinatorial kernel. General nonsense that doesn't have a combinatorial kernel feels vacuous. Then again, I'm an algebraic combinatorialist. $\endgroup$
    – Alexander Woo
    Commented Apr 13, 2021 at 0:46

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Typically, students learn about finite groups long before they think of "variety of algebras" or "compactness theorem". I suppose you also want to exclude fields from "algebra"?

I would say: topics for an algebra syllabus are chosen according to how useful they are; not according to how they fit into some general framework.

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  • $\begingroup$ in this sense fields are not 'algebraic' either - witnessed by non closure under products, division being only a partial function, etc - even if axiomatizable; but i understand a field is really studied via its modules/spaces and commutative algebras, these being (generally non finitely axiomatizable) varieties, so there's no 'problem' in this case; more broadly, do you mean it's more a question of historical and pedagogical character than of a conceptual one? $\endgroup$
    – ac15
    Commented Apr 12, 2021 at 21:09
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    $\begingroup$ Correct. Mathematics students are much more likely to use finite groups in their future studies, than they are to use varieties of algebras. $\endgroup$
    – Gerald Edgar
    Commented Apr 12, 2021 at 21:16

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