Re: A Simple Proof That The Dream Is Over

On 11 Feb 2006 18:06:26 -0800, gofab.com <tplqqq@xxxxxxx> wrote:



From: Colm K. Mulcahy (colm#NoSpam.mathcs.emory.edu)
An application of mathematics to the Beatles

A Simple Proof That The Dream Is Over

by

Colm Mulcahy


[submitted to the Journal Of Algebraic Codology]

Let G = {j, p, g, r} be a four element group, with distinguished element j.

Note: we do not (need to) assume, as most previous authors seem to have,
that j is the identity of this group.


Theorem 1: G does not exist

Proof: Conjectured since the early 70s, Chapman gave a rather convincing
proof of this in late 1980 [1], perhaps inspired by [3].

Now let G' = G \{j} - a three element set.


Theorem 2: G' is not a group.


Proof: It suffices to show that G' is not a subgroup of G. But this
follows from Lagrange's Theorem ([2], 9.1), as 3 doesn't divide 4
(see [4]).

Remarks:

(1) The author hopes that the results presented here, while not new, may
now reach a wider audience, thus laying to rest, once and for all, the
absurd assertion that {p, g, r} could form a viable group. While it is
true that three element groups exist [5], under the conditions described
above it is clear that the only way {p, g, r} could form a group would be
if the binary operation on the elements were redefined. In other words,
and this is the key point here, *no group structure exists on {p, g, r}
which is induced by the relationships which were present between the
members of the original group G *.

And it's a bit late in the day to be redefining binary operations if you ask me.

(2) In spite of over 25 years of research by scholars worldwide (eg,
[6]), the precise nature of the relationships between the four group
members of G remains shrouded in mystery. While there is convincing
evidence that the member denoted by j played the role of group identity
(indeed P. Erdos is rumoured to have proved a probabalistic result to that
effect), we should be cautious before jumping to conclusions.


References

[1] "Annihilating Operators" by Mark Chapman, Journal of Irreproducable
Results (Vol XII, No. 8, 1980)

[2] "Contemporary Abstract Algebra" by Joseph Gallian (2nd ed., 1990),
published by D. C. Heath.

[3] "Happiness Is A Warm Gun" by John Lennon, The White Album, 1968

[4] "The Ladybird Book Of Computer Assisted Arithmetic" by A. Lenstra,
A. Lenstra & H. Lenstra (London, 1985)

[5] Bruce Reznick, personal communicational (1989)

[6] "Monuments to Smithereens: Site Seeing In Liverpool" by Saki, Journal
of Suburban Archeology (Vol 9, No. 9, 1999) (preprint)



From: John Robinson (john#NoSpam.watever.waterloo.edu)

It is hard to resist responding to Mulcahy's provocative proof that The
Dream is Over [1]. I have two comments, the first somewhat tangential, but
the second strikes at the heart of Mulcahy's thesis.

1. First note that any group of less than six elements is Abelian. This
means (for instance) that j*p = p*j. Songwritership would thus appear to be
commutative - an argument maintained on artistic (and egotistic) grounds
since the 1970s (See [2] for example).

2. G = {j, p, g, r} does indeed have a subgroup under the same binary
operator - though that group only has two members. If, therefore,

j turns out not to be the identity, (1)

j is not its own inverse (2)

then one of G1 = {p, g}, G2 = {p, r} or G3 = {g, r} is a subgroup.
Therefore the dream may not be over.

Mind you, this seems pretty unlikely to me. I look forward to an
analysis of the numbered statements above.


[1] Colm Mulcahy, "A Simple Proof that the Dream is Over",
news:rec.music.beatles. 1990

[2] MPL Communications, "Wings Over America", 1975 (?)

Ah, Rags, you must have been waiting twenty years for this post!

.

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