It is often claimed that 2 is special because it is the only even prime. This is circular because "even" is defined in reference to 2. But the reasons that mod 2 is more interesting than mod n for n≠2 are reasons that 2 is special.
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"Complex manifolds have no odd-dimensional homology" is false. There may have been a true related fact that I meant to say, but I don't remember.
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Inversion in the sense of negation? But why focus on 2-ary inversion, and not 3-ary inversion (where a, wa, and w^2a are considered a triple of 3-ary inverses, for w a primitive 3rd root of unity), or whatever n-ary thing. Perhaps this is circular too…
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Inversion of elements in a group.
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A group being a monoid with a function f such that f^2(x) = x and x * f(x) = 1. But similarly, we may consider "troupes", a troupe being a monoid with an f such that f^3(x) = x and x * f(x) * f^2(x) = 1. (And so on for any n, naturally).
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So might we (I am genuinely pondering this) consider focus on groups instead of n-troupes similarly a circular sneaking in of fake specialness of 2?
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Groups arise naturally as automorphism groups. I tend to think of "group" as meaning "thing that is isomorphic to an automorphism group", and take the group axioms as a convenient way to characterize which things are groups.
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Yes, but what is an automorphism group? A collection of endomorphism tuples (m_1, m_2) such that m_1 * m_2 = m_2 * m_1 = 1, composing under the rule (m_1, m_2) * (n_1, n_2) = (m_1 * n_1, n_2 * m_2). But perhaps a 2 has been snuck in here implicitly…
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Given a set with some sort of structure on it (which necessarily includes =), its automorphism group is the set of all functions from the set to itself preserving that structure, endowed with every operation that can be naturally assigned to such sets in a constructive manner.
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By preserving that structure, I mean in the iff sense, not just that structure must be carried over in one direction.
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Also the fact that any field of characteristic 2 has -1=1, no?
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This generalizes: Any field of characteristic p has its p-th roots of unity (in its algebraic closure) all equal. (But the fact that every field already contains -1 without need for further closure is precisely the fact that the 2nd primitive root of unity is already an integer.)
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Tangentially, I've often been curious when and by who in history the realization that permutations carry parity was first discovered. I'd usually attributed it to Cauchy in the 1810s, but it seems at least implicit in Cramer's discussion of his eponymous rule 70 years earlier.
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