Song of the Week #5: Simple Finite Group (of Order Two) by the Klein Four Group

Origin/Composer: M. Salomome
Year: 2004

I first heard about this from Dustin Foley on the SUMaC group. I emailed and IM’d a bunch of people later that day about it. Then, a day later, a similar message about it came through the mn-arml list from Mitcho. What makes this song great is that its audience is pretty much only math and related majors who have taken quite a few “serious” college math classes. Most people I showed this to did not laugh much, simply because phrases like “open but too dense” doesn’t have any mathematical meaning to them. So to make this funny for everyone, I made a list of the math references in the song below:

Please visit Klein Four’s Official Site for more information about the a cappella group.

Lyrics

The path¹ of love is never smooth²
But mine’s continuous³ for you^a
You’re the upper bound⁴ in the chains⁵ of my heart
You’re my Axiom of Choice⁶, you know it’s true

But lately our relation’s⁷ not so well-defined⁸
And I just can’t function⁹ without you
I’ll prove¹⁰ my proposition¹¹ and I’m sure you’ll find
We’re a finite simple group¹² of order two¹³

I’m losing my identity¹⁴
I’m getting tensor¹⁵ every day
And without loss of generality¹⁶
I will assume that you feel the same way¹⁷

Since every time I see you, you just quotient out¹⁸
The faithful image¹⁹ that I map²⁰ into
But when we’re one-to-one²¹ you’ll see what I’m about
‘Cause we’re a finite simple group of order two

Our equivalence²² was stable²³,
A principal love bundle²⁴ sitting deep^b inside
But then you drove a wedge^c between our two-forms²⁵
Now everything is so complexified²⁶

When we first met, we simply connected²⁷
My heart was open²⁸ but too dense²⁹
Our system³⁰ was already directed³¹
To have a finite limit³², in some sense

I’m living in the kernel³³ of a rank-one map³⁴
From my domain³⁵, its image³⁶ looks so blue,
‘Cause all I see are zeroes³⁷, it’s a cruel trap^d
But we’re a finite simple group of order two

I’m not the smoothest³⁸ operator³⁹ in my class⁴⁰,
But we’re a mirror pair⁴¹, me and you,
So let’s apply forgetful functors⁴² to the past
And be a finite simple group, a finite simple group,
Let’s be a finite simple group of order two
(Oughter: “Why not three?”)^e

I’ve proved my proposition now, as you can see,
So let’s both be associative⁴³ and free⁴⁴
And by corollary⁴⁵, this shows you and I to be
Purely inseparable⁴⁶. Q. E. D.⁴⁷

Math References

¹ Path: Basically the set of points that can be described by a function, or the curve along which a path integral is taken.
² Smooth: A function that can be differentiated infinitely many times.
³ Continuous: A function without “breaks.”
⁴ Upper Bound: The upper bound is a value that can sometimes be calculated to determine the maximum possible value of a certain solution.
⁵ Chain: A totally ordered set in set theory.
⁶ Axiom of Choice: An axiom of set theory stating, in the vernacular, that one can choose an element from each set in a set of sets.
⁷ Relation: A definition relating two or more mathematical objects.
⁸ Well-defined: Mathematically defined in a logical way from base axioms, and without ambiguity.
⁹ Function: Everyone should be able to get this joke 😀
¹⁰ Prove: Logically establishing a statement to be true from previous axioms and proven statements.
¹¹ Proposition: A conjecture.
¹² Finite Simple Group: Umm…these three words take a lot to explain. Well, a group is a mathematical concept that combines a set of objects together with an operation, along with three axioms. Then, you can look at symmetries and other cool stuff in that object called a group. A finite group is simply one that has a finite set of objects defined within it. A simple group, in plain English, is the most basic type of group, similar to the prime numbers in the natural numbers.
¹³ Order Two: The order is the number of elements in a group. A simple finite group of order two is very interesting in that it is the smallest prime cyclic group, with some interesting properties.
¹⁴ Identity: A value e, such that axa^-1=e
¹⁵ Tensor: A geometric entity describing a certain space.
¹⁶ Without Loss of Generality: Often abbreviated “WLOG,” this is often used in proofs to (correctly) substitute a simpler, equivalent special case for the general case to be proven.
¹⁷ An example of a bad use of WLOG. 😀
¹⁸ Quotient Out: In a quotient group, a specific type of simple finite group, certain values are said to be “quotient out” to create the group.
¹⁹ Image: The resultant figure after a mathematical tranformation.
²⁰ Map: A function that takes each value in an object and corresponds it with another value.
²¹ One-to-One: For each x, there is one f(x), and for each f(x), there is one x.
²² Equivalence: A binary relation that is transitive, reflexive, and symmetric.
²³ Stable: A polynomial in which all the roots lie in the left half-plane, or in the open unit disc.
²⁴ Principal bundle: I know this is math, but I have no idea what it means. Check Mathworld or Wikipedia or something.
²⁵ Two-forms: Another one I don’t know.
²⁶ Complexified: A vector space that has been extended to the complex numbers.
²⁷ Simply connected: In plain terms, an object that is in one piece, and without any “holes” or “loops.”
²⁸ Open: An interval that does not include the end-point.
²⁹ Dense: Basically, a set that there are infinitely many values between any two values in the set.
³⁰ System: A group of equations to be solved simultaneously.
³¹ Directed: In graph theory, a graph where the edges have specific directions.
³² Finite limit: A limit is something in calculus…and a finite limit is a limit that is finite.
³³ Kernel: There’re a lot of definitions. Check Wikipedia.
³⁴ Rank-one map: In linear algebra, the number of rows or columns that are linearly independent.
³⁵ Domain: The “input” of a function.
³⁶ Image: I think I’ve done this already…yep. Check number 19.
³⁷ Zeroes: The roots of a function, or just the number zero.
³⁸ Smoothest: Check number 2.
³⁹ Operator: A symbol that defines a specific relationship between mathematical objects.
⁴⁰ Class: Any group of set with a certain distinguishable trait.
⁴¹ Mirror Pair: The paired point in a reflection. Or in physics, the anti-particle or supersymmetric particle of a particle.
⁴² Functors: Don’t know. Check Mathworld or Wikipedia.
⁴³ Associative. The property that, with three values a, b, and c, and operator ^, (a^b)^c=a^(b^c). Associativity is one of the axioms of groups.
⁴⁴ Free: A variable that is not dependent.
⁴⁵ Corollary: A result or statement that comes obviously from a statement that has just been proven.
⁴⁶ Purely inseperable: ???
⁴⁷ Q.E.D.: Quod Erat Demonstrandum, or “Which was to be demonstrated” in Latin. It is often used at the end of proofs as a completion.

Perhaps a Math Reference

^a You: Maybe a play on the letter u, which is often used in calculus?
^b Deep: A theorem that has many implications and goes to a core of mathematics.
^c Wedge: A mathematical shape–a triagular prism.
^d Trap: Just sounds like it could be mathy.
^e “Why not three?” Obviously, many of the propositions in this song would be false if the finite simple group became of order three. Also, this hints at a three-some. I won’t go into what that means, if you’re a poor confused soul