# Symmetric Group

Discussion in 'Undergraduate Math' started by khosaa, Feb 28, 2011.

1. ### khosaaGuest

Hi,

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).

Thank-you,
Fran

khosaa, Feb 28, 2011

2. ### Chip EasthamGuest

The symmetries of the regular n-gon (adjacency preserving) are
known as the dihedral group of order 2n (rotations and reflections),
and the notation D(n) or D(2n) varies from author to author.

However your point is correct, this is not all of S_n. If memory
serves, S_n is the symmetry group of the (n-1)-simplex, so in the
case of S_4, look at the symmetries of the regular tetrahedron.

regards, chip

Chip Eastham, Feb 28, 2011

3. ### Ken PledgerGuest

The same question came up recently with the subject "Re: etymology of
the symmetric group", and here's what I answered on the 11th January.

Ken Pledger, Feb 28, 2011
4. ### khosaaGuest

Hi Ken,

I did do a search in this group under "symmetric groups" but somehow
did not come across your response. It's kind of interesting as the
term is actually used to *define* a type of polynomial (which as I
understand it, are those homogeneous polynomials which do not change

Thanks,
Fran

khosaa, Mar 1, 2011
5. ### scaaahuGuest

I used one of the sentences to do search in this group and found it.
Sometimes Google is not that smart anymore.

I always thought symmetric group was named after symmetry.

Anyway, thanks for the answer, I learned something today.