Introduction to Abstract Algebra: Part 1
Last year I took the classes Abstract Algebra I and II from the great Joseph Gallian who literally wrote the book on abstract algebra. I loved the classes and learned a lot. This is why I decided to write this series of posts.
Part One - Introduction to Groups
Part Two - Cyclic Groups and Subgroups
Part Three - Review and the Symmetric Group
In school growing up, I learned that "algebra" was the study of equations and variables.
Wikipedia defines Algebra as the study of mathematical symbols and the rules for manipulating those symbols.
Wikipedia also mentions the interesting fact that the word "algebra" comes from the Arabic word "al-jabr" which means "reunion of broken parts" which should seem familiar if you've ever solved an equation or done any regression or "completed" the square etc...
A huge part of this study ends up being "Abstract" algebra. This refers to the study of algebraic structures such as "groups", "rings", "integral domains" ,"fields", "modules", etc... which each have more and more structure. The practical motivation for studying such structures is that they were specifically designed to model the theoretical properties of real world systems.
What is a group?
A group is a set, G, with a binary operation (lets call it *) and the following additional structure:
- Closure - if a,b are in G, then a*b is in G
- Associativity - (a*b)*c = a*(b*c)
- Identity - G must have one element, e, where e*b = b for all b in G
(i.e. a "do nothing" element) - Inverses - For every element, a, in G there must be a b in G such that a*b = e
Make sure you understand these properties firmly before trying to understand some of the rest of the material since these are fundamental to everything that follows
Associativity is often confused with Commutativity...(or at least I confuse them often)...So note that a group DOES NOT NEED TO BE COMMUTATIVE. (i.e. a*b = b*a need not be true)
If a group HAPPENS to be commutative, then it is sometimes called "abelian" after this guy.
Important Examples of Groups:
Cyclic Groups
For the sake of clarity, we will use the symbol "+" for our operation here since the operation is *basically* addition...however if you see multiplicative notation elsewhere, realize that it's an arbitrary convention and that they still mean the same thing if they're talking about cyclic groups.
Consider the set {0,1,2,3,4}.
Let "+" be addition where the result is taken modulo 5.
(This means that 0=5=10=15=..., 1=6=11=16=...,2=7=12=17=...,3=8=13=18=..., and 4=9=14=19=...)
So...how is this a group?
It's a group because it has the group properties:
- Closure - (i.e. 4+3 = 2 and you can never get a number outside {0,1,2,3,4})
- Associativity - (a+b)+c = a+(b+c) (regular addition is anyway, but check for yourself)
- Identity - 0 is a "do nothing" element or Identity for this group.
- Inverses - Every element has an inverse...(0+0=0, 1+4=0, 2+3=0)
Dihedral Groups
The plane symmetries of any regular polygon form a group called the Dihedral Group and the name for the group of symmetries of an n-gon is Dn.
Let's look at the symmetries of a square for example.
(when we talk about "Symmetries" we're talking about "rotations" and "reflections".)
You can rotate the square 0,90,180, or 270 degrees.
You can also reflect horizontally,vertically, or across either diagonal.
This means that D4 has 8 elements. (indeed Dn will always have 2n elements).
Now consider a 90 degree turn clockwise and then a horizontal flip.
This is the same as a flip (reflection) across a diagonal!
This would be written as R90*H=D1
(if R90 is the element corresponding to rotating 90 degrees, H corresponds to a horizontal flip, and D1 means reflecting across the first diagonal)
So...how is this a group?
It's a group because it has the group properties:
- Closure - Any sequence of reflections or rotations will give you back a different one that is in the group.
- Associativity - (a*b)*c = a*(b*c) (check it out for yourself)
- Identity - R0 is a "do nothing" element or Identity for this group.
- Inverses - Every flip is its own inverse and R90*R270 = R0 and R180*R180 = R0
Extra credit: Euler's formula with Introductory Group Theory
(He makes cyclic groups seem a bit more geometrical than I've seen them portrayed elsewhere, but that's not a bad thing...just try to understand how it relates to how they are traditionally described.)