- ...Cyc
^{1} - 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...,
^{2} - Usually
*D*_{n}.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...,
^{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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... commutative
^{4} - Or
*abelian*, after the mathematician Abel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... 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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .