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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?

name. Any ideas?

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?

> 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%

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.

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?

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."

See also:

http://cameroncounts.wordpress.com/2010/05/11/the-symmetric-group-3/#comment-1329

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

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This guy:

http://cameroncounts.wordpress.com/2010/04/30/the-symmetric-group-1/

knows way more than me about this. Hope it'll help.

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?

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?

> 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.

Jan 10, 2011, 3:53:18 PM1/10/11

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In article

<6147d88d-bfdc-4d36...@o14g2000prn.googlegroups.com>,

FredJeffries <fredje...@gmail.com> wrote:

<6147d88d-bfdc-4d36...@o14g2000prn.googlegroups.com>,

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.

Jan 10, 2011, 5:37:26 PM1/10/11

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> In article

> <6147d88d-bfdc-4d36...@o14g2000prn.goog

> <6147d88d-bfdc-4d36...@o14g2000prn.goog

This sounds like the original version of Galois groups,

in that the groups are defined in terms of the relations between variables that are preserved by the permutations.

Is there something to this?

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