...Cyc¹

More commonly known as C_n or Z_n, due to Z being a name for the integers. This is all part of the tendency for any field of mathematics to adopt the Roman and Greek alphabets for its own exclusive use, which is fine as long as you don't try to change fields. As I am a generalist writing for generalists, I strive for a larger namespace. Judging by the diversity of the group theory books I have looked at, choosing my own notation is no great act of rebellion.

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...,²

Usually D_n.

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...,³

Usually S_n, for "symmetric group. So why aren't I calling it Symm_n? Because then I'd have to say why it's called symmetric, and I can't. It's not obviously symmetric, and Douglas Hofstadter, who aimed at visualization and intuitive understanding, skipped this. The books mumble about "symmetric polynomials", which work, but have no obvious motivation.

But the books clearly and early define S_n as the group of permutations of n elements, so I may as well call it that.

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... commutative⁴

Or abelian, after the mathematician Abel

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... so.⁵

There's a reason I emphasize this arbitrary nature of mappings. Automorphisms are a class of mappings of a group to itself which obey certain constraints, and it can be hard to find all the automorphisms just by looking for natural operations. There is always the option of generating all possible mappings and checking each one for satisfaction of the constraints. The same holds for homomorphisms between groups: one could always set up a generic mapping and check the homomorphism equation. This isn't an ideal option, but it's there.

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