@benblumsmith.bsky.social does a lovely job talking about the objects in a proof by contradiction and how they are shown to never have existed all along. I think this proof is an interesting example of another facet of the same ontological gem.
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It's interesting you mention this one because textbook presentations of it have bothered me for a long time. The way I approach it in class is to go back to the basic form: we are trying to prove P⇒Q, so we assume P and ~Q, that is, that G/Z(G) is cyclic and G is not abelian.
Thus Z(G) is a proper subgroup, so there are nontrivial cosets, and we go from there. Textbook proofs seem to mush this all up for some reason. I dunno, writing the propositional stuff out seems to help my students or at least they're good at pretending it helps. :)
personally I think that sort of thing helps, the idea that students are supposed to be able to pick apart the propositional logic of any particular statement instantly after one lesson on truth tables is a scam
Oh yeah. I spend a lot of time in Intro to Abstract Algebra "reviewing" stuff like that. (Sometimes the gap between when they had Intro to Proofs and the Algebra class is so big, calling it a review is a bit misleading.)
I never found it difficult, but then I had a bit more than three years between my first exercises on truth tables (highschool) and this type of exercise (first year maths undergraduate).
NB I also was taught what a group is in highschool, the year before truth tables. Communist textbooks, y'all.
I was thinking about this version too, and I think it has exactly the same Weird Twist Ending: Suppose *explicitly* that x,y are non-central. Then they're *explicitly* in nontrivial cosets of Z, \ldots, haha just kidding they're central after all!!! Excuse me, what??
This reminds me of the proof that pi is transcendental.
In a way, the theorem is almost a shame, because if pi were algebraic and gamma were a "Galois conjugate" (another root of the minimal polynomial), then
e^(gamma*i)
would probably be a very interesting number. 1/
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NB I also was taught what a group is in highschool, the year before truth tables. Communist textbooks, y'all.
If G is a finite nonabelian group, then |Z(G)| ≤ |G|/4.
You have probably seen this one, but just in case you haven't, I won't spoil it.
In a way, the theorem is almost a shame, because if pi were algebraic and gamma were a "Galois conjugate" (another root of the minimal polynomial), then
e^(gamma*i)
would probably be a very interesting number. 1/