I am wondering why the group of all permutations on a set of n elements is called a "symmetric group". (Sometimes knowing the origin of a word helps me to better understand it). Neither wikipedia, nor any of the texts I own (Hungerford's "Intro to Algera", Herstein, or Dummit and Foote), directly answer this question.
I understand that the subgroup of all symmetries of a regular n-gon are contained within S_n, but it seems to me that not all the elements of S_n are in fact symmetries of a regular n-gon. (i.e. for n=4, I don't think the permutation (1324) is part of the octic subgroup of S_4).