@futurebird @johncarlosbaez Anyway, my point is that "decoding alien hieroglyphs" is actually a pretty good way to start to get an intuition for what an automorphism is.For example, if you were given all the alien hieroglyphs that correspond to their integers, and given access only to their addition table, then you'd be able to figure out what 0 is, as any identity must be unique, if it exists. Similarly you'll be able to figure out which two hieroglyphs correspond to ±1, in the sense that you could call one of them "1 up" and "1 down", and from there figure out what "2 up" and "3 down" are, but you'll never be able to decide if "up" correponds to positive and "down" corresponds to negative, or vice-versa.This intuition is captured by the fact that the group of integers under addition has exactly one non-trivial automorphism: we can negate everything everywhere and things will still work out. (And in fact, this is the only such change that is guaranteed to work perfectly in all cases.)Of course, if you then gain access to the alien's multiplication table, you can multiply "1 up" by "1 up", and that answer will correspond to the positive direction. Thus we can fully decode the alien's integers, which corresponds to the fact that the ring of integers exhibits only the identity automorphism: when multiplication is involved, we can't just flip everything's sign and expect things to work out.This intuition is a bit hard to operationalize, though, as the addition tables are infinitely large. In reality, if the alien heiroglyphs are truly capable of expressing arbitrarily large members of an infinite set, such as the rational numbers, the notation must involve some regularity. That regularity can provide insight into the alien's interpretation of their rationals in ways that don't correspond to what could be learned from their operation tables alone.The automorphisms of the group of rationals under addition correspond to multiplying by a non-zero rational number, capturing the intuition that you'll never be able to definitively decode the scale of the alien's unit of measurement from the addition table alone. But if you see something like 1/10000, you can guess it's probably not the unit, versus something much simpler like 1/1.However, the field of rationals exhibit only the trivial automorphism, meaning that you could fully decode alien rationals from their addition and multiplication table.Switching to a finite system avoids these complications, and also is capable of providing much more interesting examples of automorphism groups than your more widely-appreicated number systems can.
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