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Cam McLeman's user avatar
Cam McLeman's user avatar
Cam McLeman's user avatar
Cam McLeman
  • Member for 14 years, 11 months
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Isomorphism and number of subgroups
(non-isomorphic groups of the same order, that is...)
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Isomorphism and number of subgroups
You have the logic backwards. If there can be non-isomorphic groups with a structure-preserving bijection between their subgroups, certainly there can be non-isomorphic groups with the same number of subgroups.
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Isomorphism and number of subgroups
I just noticed that you yourself posted this link in the other question. Why doesn't this answer your question completely?
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Isomorphism and number of subgroups
A much stronger question was asked and answered (negatively!) here: mathoverflow.net/questions/35455/…
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How to describe a tree? (depth, degree, balance, ... what else?)
There are many many adjectives which describe trees. I'd recommend you spend some time with a nice survey article (or maybe even the wikipedia page) to get acquainted with them. Your question about how balanced a tree is sounds potentially interesting, but I think you'll be able to ask it better after having done some more reading.
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Why aren't there more classifying spaces in number theory?
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Why aren't there more classifying spaces in number theory?
+1: Thanks for this interpretation -- I think this makes a lot of sense, though I'm going to spend some time working it out more carefully. Do you have a reference for the historical comment? I'd like to understand the historical development better.
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Please check my 6-line proof of Fermat's Last Theorem.
So certainly a lot of these techniques become hard/impossible as you move away from quadratic Diohpantine equations, right? I'm thinking methods 1, 2, and 7 in particular -- and 3 gets harder and harder as well. Maybe this should just be an exercise for myself, but it is clear to you that it would be rather difficult to find examples where the Hilbert Class Field approach would the best/only one?
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Why aren't there more classifying spaces in number theory?
@Agol: Well, perhaps not directly, but it sounds fascinating regardless, so +1. If there's an answer there to be elaborated on, I'd love to see it. This will save me from asking the new question "What was Agol talking about when he said..." :)
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Why aren't there more classifying spaces in number theory?
@Dev Sinha: Okay, fair point. On the other hand, there are a lot of important statements in group cohomology which involve trivial coefficients: The Schur multiplier (or Hopf's Integral Homology Formula) comes to mind, as does the interpretation of the relation rank of a pro-$p$-group $G$ as $\operatorname{dim}H^2(G,\mathbb{F}_p)$.
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How many different representations of pi can we come up with?
Try explaining that to the health inspectors...
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What category without initial object do you care about?
But if you do that, then this comment will look really out of place...
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Dividing a square into 5 equal squares
Very nice answer! Will have to remember this next time I teach number theory.
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quiver mutation
Perhaps you should include your few known applications in the question. No sense in having someone do the work to type up ones you already know.
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