Re: Need a Name for group operations

In article
<14518551.1176226441320.JavaMail.jakarta@xxxxxxxxxxxxx
forum.org>,
PaulHjelmstad <phjelmstad@xxxxxxx> wrote:

On Apr 10, 9:23 am, PaulHjelmstad
<phjelms...@xxxxxxx> wrote:
With some of the work I am doing in music
theory
(for
a paper I want to write), I need to consider
musical
sets in Z12 that map into themselves under D4 X
S3.

I want to break this down to consider sets that
map
into themselves under D4 X C3, but only for D4
permutations that go backwards (that is,
(24),(14)(23), (13),(12)(34)) Is there a name for
this?

The same applies to S3 X C4, I only want the
backwards
permutations of S3.

What, exactly, do you mean by a "backwards"
permutation? Don't just
provide examples; what is your /definition/ of a
backwards
permutation?

Cheers - Chas

Yes this is tricky. Because, (1432) is NOT what I
would
define as a backwards permutation, because it
leaves
the square as

1234
4123

and this is forwards. What I mean is the four
permutations
of D4 that reverse the order of the elements, with
the example given. Same
with S3.

I don't know what to add, but here is the results
of
the above perms,

1432
4321
3214
2143

For S3

132
321
213

If I had a good definition, I'd probably have the
name too!

If I understand what you're doing (and that's a big
if), to you D_4
is a subgroup of S_4. Well, if I'm not mistaken, S_4
has three subgroups
isomorphic to D_4. Is one of them somehow
distinguished for you?

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for
email)


Thanks. Yes, that's it, but it's worse, I am just considering 1234, 2341, 3412, 4123, 1432, 4321, 3214, 2142 (Symmetry of the Square) and taking only those
that go around the square backwards in a group
direct product with C3. This is to characterize what
is called the M5 relation in music. Also it is
interesting that my "D4-backwards" operation is the
exact inverse of the "S3-backwards" operation in Z12.
(The sets also get transposed but this is easy to deal with).

Example

0,1,2,3,5 -> 0,7,2,9,11 normal M5, or my "D4-backwards"
0,1,2,3,5 -> 0,5,10,3,1 inverse(M5) or m "S3-backwards"

The pitfalls of doing things on my own....

Paul
.

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