Groups
Since I’ve enjoyed the recent mathematical discussions on my friends‘ blogs, I have decided to reciprocate by writing my own exciting post about math!
Today’s topic is about the fundamental mathematical structure of abstract algebra… the group. A group is defined as a pairing (G,*) where G is a set and * is a binary operator (a binary operator is function like “+” which takes two objects and produces a third object of the same type, like 2+3=5). We usually call the group’s operation “multiplication” and denote it the same way standard multiplication is denoted. The set must satisfy the following group axioms…
- Closure under the product — if a and b are in G, then a*b must be in G.
- Identity element — there must exist an element e in G such that a*e = e*a = a for all elements a in G.
- Inverse elements – for each a in G, there must exist an element a-1 such that a*a-1 = e.
- Associativity — for all elements a, b, c in G, we must have (a*b)*c=a*(b*c).
An example of a group would be (G,+) where G = {0, 1, -1, 2, -2, 3, -3, …} is the set of all integers. The group “multiplication” is addition. The set is closed under addition (the sum of two integers is an integer), so axiom 1 is satisfied. Addition is associative, so axiom 4 is satisfied. The number 0 functions as an identity (a+0=0+a=a), so axiom 2 is satisfied. For any number a, we can find an inverse -a, so that a+(-a)=0, so axiom 3 is satisfied.
This is an infinite group (since G is an infinite set). Can you think of any finite groups?
Posted: November 12th, 2007 under Engineering.
Comments
Comment from Jed
Time 2007/11/13 at 11:25 am
Yes! Nicely done! You’re on your way to being a mathematician.
Actually, more generally, the set consisting of the identity G={e} is always a group (it is called a trivial group).
Comment from Josiah
Time 2007/11/13 at 11:26 am
At first, I thought that (G, [addition]) where G = {0,1} would be a group…
Closure: 0+1 = 1; 0+0 = 0
Identity: 1+0 = 0+1 = 1; 0+0 = 0
Inverse: 1-1 = 0; 1-0 = 1
Associative: 1+(1+0) = (1+1)+0
But, it fails at the inverse property: 0-1 = -1, which is not in the group. These things are very unusual.
Comment from Jed
Time 2007/11/13 at 11:33 am
Inverses don’t necessarily have to be produced by the “inverse operation.” For example, you could just say 0 is the inverse of 1 and 1 is the inverse of zero (but that still wouldn’t make this particular example work as a group).
Things might work out better for this group if you try addition mod 2 as the operation.
Comment from jeff
Time 2007/11/14 at 8:24 am
Very interesting. Kind of like a good Sudoku puzzle, but not really.
I’m curious about this statement: “We usually call the group’s operation ‘multiplication’ and denote it the same way standard multiplication is denoted.”
That terminology seems confusing. Why did the mathematicians choose to resuse a word with existing meaning, rather than apply a different perfectly good word (such as “operator”) that could avoid any confusion? After all, you’d think mathematicians would do a better job than English majors, who brought us such things as “fair”, “fair”, and “fare”.
So, how do you feel about groups? What does it mean to you? Has its existence been useful to you in any way?
Comment from Jed
Time 2007/11/14 at 11:27 am
Yep, a Sudoku puzzle looks very much like a group multiplication table (since every Sudoku puzzle forms a Latin square), but the requirement that each of those nine boxes contain the numbers 1 through 9 messes things up (it prevents us from forming inverses).
I don’t really know where that type of mathematical notation comes from! I think it probably arose when mathematicians began shortening “a*b” to “ab”. Also, the more “new” terminology that gets introduced, the weirder the wording becomes (we end up with terms like “meager”, “homeomorphism”, and “abelian”)! So sometimes it’s better to just re-use wording!
Haha… what do groups mean to me? Absolutely nothing. They represent an abstract concept whose implications I have not yet been able to visualize. Which is rather unfortunate.
Comment from Eloprah
Time 2007/11/18 at 6:45 pm
I’m going to comment here….not because it’s about the topic, but because it’s not. You need a place to comment random comments.
Anyways, I like the picture on the top of the screen. Who drew it?
Comment from Jed
Time 2007/11/22 at 9:49 am
Hi Eloprah! Your wish has been granted… I have added a page called “Chat” (there’s a link on the left side of the page) to facilitate random comments!
I’m glad you like the picture. The artist was a certain J. R. McClurg.
Comment from Josiah
Time 2007/11/13 at 11:19 am
Wouldn’t the Group (G, [multiplication]) where G = {1} be a finite group?
Closure: 1*1 = 1
Identity: 1*1 = 1
Inverse: 1*(1-1) = 1
Associative: 1*(1*1) = 1*1*(1) = 1