## Monday, April 17, 2006

### Finite Groups and the Greatest Math Song Ever!

I'm busy tonight, and in a strange mood, so I'm not going to say too much; but I need to get to one point, so that I can link to something wonderful.

So, yesterday, I posted an introduction to group theory.

Well, it turns out (naturally), that there are many groups with finite numbers of members. And they are, quite imaginatively, called finite groups. Most finite groups can be illustrated quite beautifully with a diagram, but that's a topic for another day.

Within a finite group, the number of elements in the group is called the order of the group. So, for example, a group with 10 elements is called a finite group of order 10.

There also a notion, too complex to really get into tonight, called a simple group. A simple group is, essentially, one that has a kind of minimality about it; its only normal subgroups (basically groups formed by subsets of the values in the group) are trivial. (Note: the word "normal" was originally omitted from this sentence, because I forgot that there are plenty of non-simple subgroups; only the normal ones need to be trivial. This error was pointed out by commenter Ivan M.)

So - the point of this: here is a link to the greatest a-capella math song of all time: Finite Simple Group of Order Two, by the Klein 4 at Northwestern University.

Yes, I am a dork.

• Prove your dorkiness...post a representation of the monster. :)

By  A, at 10:08 PM

• 5 meg wav !
MP3 people !

By  Anonymous, at 12:05 AM

• Wave file? It's a WMV.

By  Thomas Winwood, at 1:56 AM

• I suppose this song is one notch of sophistication above Tom Lehrer's "Derivative Song" and "There's a Delta for Every Epsilon".

(video at http://www.archive.org/details/lehrer)

By  Blake Stacey, at 6:00 AM

• anonymous:

I would prefer to have a smaller mp3 of the song that the great big windows-media file with video; but it's not mine! I don't have the right to convert it to a different format and then post it on my site.

-Mark

By  MarkCC, at 8:11 AM

• Actually, you got the definition of simple group wrong. A simple group can (and usually does) have non-trivial subgroups. It's simple if it has no non-trivial normal subgroups.

The finite simple group are analogous to prime numbers, and the the Jordan-Holder theorem (a group's composition series is unique) is group theory's analog to the Fundamental Theorem of Arithmetic (the prime factorization of a number is unique).

By  Ray, at 9:31 AM

• Get Dawn Upshaw & a Schubertesque (or Wolf perhaps) setting of this Lem poem, & then you would have the greatest math song:

http://www.ee.duke.edu/~wrankin/misc/tensor.html

Come, let us hasten to a higher plane
Their indices bedecked from one to n
Commingled in an endless Markov chain!

Come, every frustrum longs to be a cone
And every vector dreams of matrices.
Hark to the gentle gradient of the breeze:
It whispers of a more ergodic zone.

In Riemann, Hilbert or in Banach space
Let superscripts and subscripts go their ways.
Our asymptotes no longer out of phase,
We shall encounter, counting, face to face.

Thou'lt tell me all the constants of thy love;
And so we two shall all love's lemmas prove,
And in our bound partition never part.

For what did Cauchy know, or Christoffel,
Or Fourier, or any Bools or Euler,
Wielding their compasses, their pens and rulers,
Of thy supernal sinusoidal spell?

Cancel me not - for what then shall remain?
Abscissas some mantissas, modules, modes,
A root or two, a torus and a node:
The inverse of my verse, a null domain.

Ellipse of bliss, converge, O lips divine!
the product o four scalars is defines!
Cyberiad draws nigh, and the skew mind
Cuts capers like a happy haversine.

I see the eigenvalue in thine eye,
I hear the tender tensor in thy sigh.
Bernoulli would have been content to die,
Had he but known such a^2 cos 2 phi!

By  steve, at 2:09 PM

• Gosh it's nice to see the spirit of Tom Lehrer still lives! These guys are great.

By  AndyS, at 2:41 AM

• Great poem, Steve. One complaint. "Euler" is prounounced "Oil-er". It's German. However, since in English we rhyme "again" with "rain"... We'll have to call it poetic license.

By  Qalmlea, at 9:26 AM

• Gamela said:
Great poem, Steve. One complaint. "Euler" is prounounced "Oil-er". It's German. However, since in English we rhyme "again" with "rain"... We'll have to call it poetic license.

The poem is from Stanislaw Lem's book The Cyberiad. Considering it was translated from the original Polish
I am not sure if the Licence was Lem's or the tranlsator's

By  steve, at 11:05 AM