What is a "group" in Maths? and why proofs that 1=2 are bogus
Created | Updated Mar 16, 2002
Preamble:
Mathematical terms have a variety of (well, two) origins.
1) Mathematicians take a word that everyone knows, like "ring" or "map" and decide it means something else. Physicists, of course, are worse than mathematicians in this respect, as they will then go on to insist that anyone using the commandeered term in its everyday use is now wrong (e.g. "That's not its weight; it's its MASS." and "That's not a force, it's just a demonstration of the centrifugal effect.").
2) They make something up.
The term "Group" falls into the first category.
In order to know what a group is, you must first understand these mathematical word-corruptions:
a) a "set" means some stuff written between curly {} brackets;
b) a "binary operation" on a set A is some way of specifying a particular element (thing which is in a set) of A for any particular pair of elements of A. For instance, the operation of multiplication gives you a particular integer (e.g. 42) for any particular pair of integers (e.g. 6 and 9).
Definition (Mathematicians like definitions a lot):
A group consists of a set (denoted G) and a binary operation (denoted *) on G that satisfy three conditions:
1) associativity;
2) the presence of a neutral element;
3) the presence of inverses.
_Associativity_ means if you take any three elements of G (call them a, b and c), then a*(b*c)=(a*b)*c.
So if you're dealing with addition, it means that if you have 2,7 and -4:
2 + (7 + -4) = 2 + 3 = 5
and
(2 + 7) + -4 = 9 + -4 = 5
As this works for any three integers, addition is associative on the set of integers.
A _neutral element_ is an element which doesn't really do anything. If our group has a neutral element e (which must be an element of G) then for any other element a in G:
e*g=g*e=g
So in addition on integers, 0 is the neutral element, because
0+n=n+0=n for any integer n (try it, it really works).
Finally, inverses:
This means that for each element of G, there's another element you can pair it with that produces the neutral element.
For any a in G, there is some b in G such that a*b=b*a=e.
(--In fact, it's straightforward to show that inverses are unique: any element can have only one inverse in a group. If you're not bothered, skip ahead to the next bit.
Suppose b and c are inverses of a (so a*b=e and a*c=e).
Then a*b=a*c.
Therefore b*(a*b)=b*(a*c)
By associativity: (b*a)*b=b*(a*b) and (b*a)*c=b*(a*c)
therefore (b*a)*b=(b*a)*c = 1*b = 1*c
so b=c, and the "two" inverses must be the same thing.
Thank you.
--)
In the case of addition, the inverse of an integer a is written -a, so that
a + -a = -a + a = 0. In multiplation inverse elements are naturally enough called "inverses", and the multiplicative inverse of a can be written 1/a. This works because the neutral element for multiplication is 1, so a*1/a=(1/a)*a=1.
This brings up the reason addition of integers qualifies as a group, but multiplication of integers DOES NOT.
1) Firstly, the inverses of most integers are not integers (one half is the inverse of 2, but one half is not an integer). In fact, in order to make the integers into a group for multiplication we need to included anything you can get by multiplying integers and taking inverses. This is what provides us with the set of rational numbers, wherein any element (written a/b) has its inverse in the element b/a.
2) Secondly, however the rational numbers also fail to be a multiplicative group because of that pesky zero, which was so useful in addition. Since multiplying by zero always produces zero, it cannot have an inverse.
Digression:
Suppose you introduce an element # to serve as the inverse of zero, defining its multiplication in some way such that 0*#=#*0=1. You then have to decide what multiplying by # does to all the other numbers lying around in your set. By associativity, (2*0)*# must equal 2*(0*#), which equals 2*1=2.
However, 2*0=0, so (2*0)*#=0*#=1 by definition. This, in variously simplified and more complicated terms, is the basis of stupid people's claims to have proven that 1=2. In maths it's called "Proof by contradiction", and it amounts to mathematical sarcasm. It's obvious that 1 does not equal 2, so some premise that produced this result must be false: in this case, 0 cannot have a multiplicative inverse, because that would mean 1=2.
So neither integers nor rationals qualify as groups under multiplication, though both are groups under addition. To get a multiplicative group you simply exclude zero: the nonzero rationals are a group under multiplication.
More examples of groups:
-The set {1} under multiplication: the only pair the operation can apply to in this case is 1*1=1, so we have the inverse and the neutral element all in one number. Similarly, the set {0} under addition. (Mathematicians consider these moreorless the same group because they're basically doing the same thing but with different names).
-The sets of real numbers, complex numbers, rationals, integers, even numbers, or multiples of any number (e.g. multiples of 3, or even multiples of 1/2 or of pi) under addition, with neutral element 0.
-The sets of nonzero real numbers, nonzero complex numbers, and nonzero rationals under multiplication with neutral element 1.
The fact that you can find the same structure with so many sets and operations is what makes mathematicians get all excited and want to start defining things. Once defined, groups can be found all over the place, in sets of functions, sets of vectors, sets of words, sets of sets, and sets of strange things no-one has a word for. And as soon as they find a group, they can instantly deduce bucketloads of properties about it, like isomorphism theorems, and ... probably other things as well. This is how maths works - by getting as far away from the real world as possible and seeing what you find, on the pretext that you'll manage at some point to apply what you've found to something more specific in the real world, given just a few obscure properties.
Mathematical terms have a variety of (well, two) origins.
1) Mathematicians take a word that everyone knows, like "ring" or "map" and decide it means something else. Physicists, of course, are worse than mathematicians in this respect, as they will then go on to insist that anyone using the commandeered term in its everyday use is now wrong (e.g. "That's not its weight; it's its MASS." and "That's not a force, it's just a demonstration of the centrifugal effect.").
2) They make something up.
The term "Group" falls into the first category.
In order to know what a group is, you must first understand these mathematical word-corruptions:
a) a "set" means some stuff written between curly {} brackets;
b) a "binary operation" on a set A is some way of specifying a particular element (thing which is in a set) of A for any particular pair of elements of A. For instance, the operation of multiplication gives you a particular integer (e.g. 42) for any particular pair of integers (e.g. 6 and 9).
Definition (Mathematicians like definitions a lot):
A group consists of a set (denoted G) and a binary operation (denoted *) on G that satisfy three conditions:
1) associativity;
2) the presence of a neutral element;
3) the presence of inverses.
_Associativity_ means if you take any three elements of G (call them a, b and c), then a*(b*c)=(a*b)*c.
So if you're dealing with addition, it means that if you have 2,7 and -4:
2 + (7 + -4) = 2 + 3 = 5
and
(2 + 7) + -4 = 9 + -4 = 5
As this works for any three integers, addition is associative on the set of integers.
A _neutral element_ is an element which doesn't really do anything. If our group has a neutral element e (which must be an element of G) then for any other element a in G:
e*g=g*e=g
So in addition on integers, 0 is the neutral element, because
0+n=n+0=n for any integer n (try it, it really works).
Finally, inverses:
This means that for each element of G, there's another element you can pair it with that produces the neutral element.
For any a in G, there is some b in G such that a*b=b*a=e.
(--In fact, it's straightforward to show that inverses are unique: any element can have only one inverse in a group. If you're not bothered, skip ahead to the next bit.
Suppose b and c are inverses of a (so a*b=e and a*c=e).
Then a*b=a*c.
Therefore b*(a*b)=b*(a*c)
By associativity: (b*a)*b=b*(a*b) and (b*a)*c=b*(a*c)
therefore (b*a)*b=(b*a)*c = 1*b = 1*c
so b=c, and the "two" inverses must be the same thing.
Thank you.
--)
In the case of addition, the inverse of an integer a is written -a, so that
a + -a = -a + a = 0. In multiplation inverse elements are naturally enough called "inverses", and the multiplicative inverse of a can be written 1/a. This works because the neutral element for multiplication is 1, so a*1/a=(1/a)*a=1.
This brings up the reason addition of integers qualifies as a group, but multiplication of integers DOES NOT.
1) Firstly, the inverses of most integers are not integers (one half is the inverse of 2, but one half is not an integer). In fact, in order to make the integers into a group for multiplication we need to included anything you can get by multiplying integers and taking inverses. This is what provides us with the set of rational numbers, wherein any element (written a/b) has its inverse in the element b/a.
2) Secondly, however the rational numbers also fail to be a multiplicative group because of that pesky zero, which was so useful in addition. Since multiplying by zero always produces zero, it cannot have an inverse.
Digression:
Suppose you introduce an element # to serve as the inverse of zero, defining its multiplication in some way such that 0*#=#*0=1. You then have to decide what multiplying by # does to all the other numbers lying around in your set. By associativity, (2*0)*# must equal 2*(0*#), which equals 2*1=2.
However, 2*0=0, so (2*0)*#=0*#=1 by definition. This, in variously simplified and more complicated terms, is the basis of stupid people's claims to have proven that 1=2. In maths it's called "Proof by contradiction", and it amounts to mathematical sarcasm. It's obvious that 1 does not equal 2, so some premise that produced this result must be false: in this case, 0 cannot have a multiplicative inverse, because that would mean 1=2.
So neither integers nor rationals qualify as groups under multiplication, though both are groups under addition. To get a multiplicative group you simply exclude zero: the nonzero rationals are a group under multiplication.
More examples of groups:
-The set {1} under multiplication: the only pair the operation can apply to in this case is 1*1=1, so we have the inverse and the neutral element all in one number. Similarly, the set {0} under addition. (Mathematicians consider these moreorless the same group because they're basically doing the same thing but with different names).
-The sets of real numbers, complex numbers, rationals, integers, even numbers, or multiples of any number (e.g. multiples of 3, or even multiples of 1/2 or of pi) under addition, with neutral element 0.
-The sets of nonzero real numbers, nonzero complex numbers, and nonzero rationals under multiplication with neutral element 1.
The fact that you can find the same structure with so many sets and operations is what makes mathematicians get all excited and want to start defining things. Once defined, groups can be found all over the place, in sets of functions, sets of vectors, sets of words, sets of sets, and sets of strange things no-one has a word for. And as soon as they find a group, they can instantly deduce bucketloads of properties about it, like isomorphism theorems, and ... probably other things as well. This is how maths works - by getting as far away from the real world as possible and seeing what you find, on the pretext that you'll manage at some point to apply what you've found to something more specific in the real world, given just a few obscure properties.
