An Introduction To Group Theory

Part Of: Algebra sequence
Content Summary: 900 words, 9 min read

Rediscovering Addition

Do you remember how, when learning addition in elementary school, you were taught the Associative Rule and the Commutative Rule?

It turns out that those two ideas were summarizations of five axioms (recall that Ƶ represents the set of all integers { … , -2, -1, 0, 1, 2, … } ):

Group Theory- Addition

Take a minute to prove to yourself that the above axioms encapsulate all of arithmetic.

If we think of the above as a function, three inputs are relevant to us: the target set, the operator, and the identity element. Thus, we can condense the above into:

(Ƶ, +, 0)

Another interesting observation: the axiom for Inverse Element motivates subtraction.

Rediscovering Multiplication

Do you remember how, when learning multiplication in elementary school, you were taught the Associative Rule and the Commutative Rule?

It turns out that those two ideas were summarizations of five axioms:

Group Theory- Multiplication (1)

Take a minute to prove to yourself that the above axioms encapsulate the entirety of multiplication.

If we think of the above as a function, three inputs are relevant to us: the target set, the operator, and the identity element. Thus, we can condense the above into:

(Ƶ, *, 1)

Another interesting observation: the axiom for Inverse Element motivates division.

Generalized Algebraic Structures

Did the above two sections feel painfully similar? I thought so.

One lesson they teach you in computer science is: if you notice yourself copy-pasting code, you should try to consolidate your software into one function.

Analogously, we can generalize addition and multiplication, like so:

Group Theory- Groups Generalize Addition And Multiplication

Here, we generalize our three inputs defined above:

  • Z becomes a specific instance of an input set
  • + and * become specific instances of a general class of operator.
  • 0 and 1 become specific instances of a general class of identity elements.

Let us call this particular set of five axioms an Abelian group (also known as a commutative group).

Group Theory- Abelian Groups As A Function

Axioms Deserve The Knife

I like to think of math as games with formal systems. Well, let’s play a game called “trim the axiom on the group”:

Group Theory- Trimming Abelian Groups

Of course, there is no particularly strong reason to “knife” axioms in this order. More esoteric options are available (in the below, red axioms are removed, green are re-introduced).

Abelian- Other Group TypesA couple complaints:

  • These naming conventions feel a bit convoluted.
  • The “different strategies” don’t feel comprehensive.

A diagram of the entire state-space of groups address these complaints:

Group Theory- Group Space

I like this graph for several reasons:

  • A new, more efficient, taxonomic system could easily be devised.
  • This is the first graph whose complexity begins to suggest “I wonder if I could compute over an axiomatic state space”.

Generalizing Group Marriage

The above taxonomy provides a nice language by which to discuss algebraic structures. But our elementary school compatriots are still able to do something we cannot: they are able to both add and multiply within the same algebraic structure.

Let us marry addition-groups and multiplication-groups together, into a field:

Group Theory- Group Marriage (1)

Why does the multiplicative Abelian group exclude the element zero? Because the inverse element is incompatible with that single element of the set.

But just as group structures don’t have to be Abelian, we can take a razor to fields, as well:

Group Theory- Trimming Fields

Each type of group-marriage has its own properties; no one system is always a superior tool than its neighbors.

Concluding Musings

Takeaways:

  • Multiplication and addition share a remarkable number of properties, including (but not limited to) associativity, commutativity, and an identity element.
  • If we define these commonalities as Abelian groups, which are algebraic structures that (at the least) accept an input set, and an operator.
  • There is no commandment that groups must satisfy all five axioms; indeed, we can identity and name 32 such group possibilities.
  • Real algebraic systems typically marry together an addition-group, and a multiplication-group; let us call these megastructures fields, or rings, depending on their component group-type.

A couple really cool things you can do to these group theoretic constructs:

  • While we motivated input sets as being numbers, the sets can truly be anything. You could apply group theory to set of matrices, or vectors, or even polynomials.
  • We started this process by taking as input a set of numbers, and outputting an algebraic system (either consistent, or not). But we can reverse this process, and ultimately define say, the Reals, as that set thing which, along with satisfying other things, magically creates a consistent field. 
  • As I hope to explore later, graphical models tend to operate on algebraic structures that are commutative semi-rings.

I have a few open questions, including:

  • Can’t we play a “hack and slash” game on a different set of axioms? Why haven’t different axioms (such as alternative algebras, a weaker cousin of associative algebras) been explored in comparable depth?
  • Can rings/fields operate against more operators then just multiplication and addition?
  • Can rings/fields be defined against more than two operators, or should we speak of “ring combinations” instead?
  • Can rings/fields be defined as the combination of groups whose input sets are dissimilar in nontrivial ways?
  • Can I create a map of inter-axiom relationships? To use the language of the axiomatic state space, is there any axiom combination (besides 00000) which is incoherent?
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