# in what sense is the subject of finite group theory 'algebraic'?

[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?

• 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. Apr 12, 2021 at 21:22
• Oh my--when would an undergraduate, or even graduate student, be allowed to read about the classification of finite simple groups? Apr 12, 2021 at 21:27
• 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
– ac15
Apr 12, 2021 at 21:28
• @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
– ac15
Apr 12, 2021 at 21:39
• @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. Apr 13, 2021 at 0:46