# etymology of the symmetric group

14 views

### Edward Green

Dec 31, 2010, 9:30:25 PM12/31/10
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I've found out what it is, but I haven't found out how it got its
name. Any ideas?

### spudnik

Jan 1, 2011, 2:51:04 PM1/1/11
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Galois?... Cauchy?... Gauss, a Golden Braid?

Jan 1, 2011, 7:45:36 PM1/1/11
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> I've found out what it is, but I haven't found out
> how it got its
> name. Any ideas?

I think S_n was originally considered to be the
group of "symmetries" (rotation, translation, etc.)
of an n-gon, and the elements of S_n were the images
of the vertices under such symmetries. But I am not
100%

### Edward Green

Jan 2, 2011, 3:55:08 PM1/2/11
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But surely one cannot achieve an arbitrary permutation of the vertices
of an n-gon by executing the symmetry operations? There is no way to
exchange two, and only two, vertices, for example.

### FredJeffries

Jan 2, 2011, 5:40:16 PM1/2/11
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On Dec 31 2010, 6:30 pm, Edward Green <spamspamsp...@netzero.com>
wrote:

> I've found out what it is, but I haven't found out how it got its
> name. Any ideas?

Oxford English Dictionary traces it to William Burnside's "Theory of
groups of finite order" in 1897:
http://www.latexnical.com/library/Burnside/William/TheoryOfGroupsOfFiniteOrder.pdf

see page 180 (200 of the pdf) where footnote reads:
"The symmetric group has been so called because the only functions of
the n symbols
which are unaltered by all the permutations of the group are the
symmetric functions."

Jan 2, 2011, 5:54:01 PM1/2/11
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This guy:

### spudnik

Jan 3, 2011, 4:34:18 PM1/3/11
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exchanging *any* two vertices wouldn't be considered
to be an elementary symmetry; eh?

Jan 3, 2011, 4:52:16 PM1/3/11
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> exchanging *any* two vertices wouldn't be considered
> to be an elementary symmetry; eh?

Yes, my bad. I spoke too soon. Fred Jeffries post
and my second post (the link) are better explanations
than the one I gave in my first post.

### Ken Pledger

Jan 10, 2011, 3:53:18 PM1/10/11
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In article
FredJeffries <fredje...@gmail.com> wrote:

> On Dec 31 2010, 6:30 pm, Edward Green <spamspamsp...@netzero.com>
> wrote:
> > I've found out what it is, but I haven't found out how it got its
> > name. Any ideas?

> ....

> see page 180 (200 of the pdf) where footnote reads:
> "The symmetric group has been so called because the only functions of
> the n symbols
> which are unaltered by all the permutations of the group are the

> symmetric functions."....

Yes. In the 19th century permutation groups were closely associated
with functions of several variables.
For example, ((x_2)^2)(x_1 + x_3) is invariant under two permutations
of its variables' subscripts: (1) and (1,3). Hence {(1), (1,3)} is
the group of the function. OTOH a given permutation group can have an
appropriate function constructed for it. Such techniques were
developed from Lagrange's work by Galois, Cauchy, Jordan, etc.

A *symmetric* function such as x_1 + x_2 + x_3 is invariant under
all permutations of its variables, so its group comprises all those
permutations. Hence the name "symmetric group".

An *alternating* function such as (x_1 - x_2)(x_1 - x_3)(x_2 - x_3)
is so called because of alternating signs: the function is invariant
under some permutations (the even ones) but has its sign changed by
others (the odd ones). Hence the name "alternating group" for the set
of all even permutations.

Ken Pledger.