The phrase “the permutation group Sn” refers to a
particular group of permutations that act on an
n -element set N— namely, all of them. For a given n ,
there are, in general, many permutation groups that
act on N. All but one are smaller than Sn .
In other words, the phrase “the permutation group Sn”
does not imply that “Sn ” is a symbol for a structure
associated with n called “the permutation group.”
It is instead a symbol for “the symmetric group,” the largest
of (in general) many permutation groups that act on N.
This point seems to have escaped John Baez.
For two misuses by Baez of the phrase “permutation group” at the
n-Category Café, see “A Wrinkle in the Mathematical Universe”
and “Re: A Wrinkle…” —
“There is such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle