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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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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Admittedly what I said doesn't apply to automorphism groups of objects in non-concrete categories. I think of these as "things that work in much the same way that sets with structure do".
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Alright, now the 2 comes from your focus on "functions"; that is, on particular 2-ary relations. But we might also look at 3-ary correspondence relations. E.g., R(x, y, z), such that for any choice of x, y, or z, there are designated corresponding values for the other arguments.
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