Section 8.3 Essential Group Facts for Number Theory
Subsection 8.3.1 Step-by-step notions to the definition
Subsubsection 8.3.1.1 Sets
Sets are just what you think. They are collections of (mathematical) stuff. In our uses of groups, we will exclusively be concerned with sets that are collections of numbers, likeSubsubsection 8.3.1.2 Binary operations
A binary operation is a set with a multiplication table on it. That's it. Usually books call itUsually this would be (say) normal addition or multiplication on numbers, though it could also be subtraction.
On the other hand, if
is the set of continuous functions on the operation could be composition of functions,
Subsubsection 8.3.1.3 Closed operations
A binary operation is called closed if you don't get anything outside the set with your operation. This is important because it's easy to break this.If you are adding two positive numbers, for instance, you always get a new positive number.
Is this still true if you subtract two positive numbers from each other?
This also can happen with division, right? You have to look at
and then you have to be careful because of the previous point.For a more complicated example, let
be the set of 2x2 matrices with determinant 1; if you add two of them, your determinant might change a lot.On the other hand, if you multiply two such matrices, you're golden; the determinant will still be 1.
Subsubsection 8.3.1.4 Associative operations
An operation is associative if it doesn't matter how you put parentheses in. This is not an algebra course, so I won't harp on this – everything we do will satisfy it in obvious ways. But it's worth noting that exponentiation is not associative, so it's not a trivial condition.Example 8.3.1.
Subsubsection 8.3.1.5 Identity
Much more important is whether your operation has an identity element. You have seen this many times before in addition and multiplication.That is,
for anyThe identity matrix under matrix multiplication is another example.
By the way, if there is an identity, there's only one, which is sometimes useful to know.
Example 8.3.2.
Here is a more interesting example. Let your set be the set of all rotations of a square which leave it facing the same way. That is, rotation by 90 degrees to the left, 180 degrees right, etc. (Think of a child's block sorter.)
The binary operation combining two (possibly different) rotations would be to first do one rotation, and then the other one.
Then an identity element
of this is just to leave the block alone!
This is sort of weird at first, but an extremely important example.
Subsubsection 8.3.1.6 Inverses
Almost there! Let's keep thinking about that last example. Say I turn the block 90 degrees to the right, then I realize I made a horrible mistake and want to get back to the original position. Is there anything I can do, short of buying a new square block? Of course there is! Just turn it back 90 degrees to the left. So if I call the first moveThe absolute prototype of this is negative numbers. That is, for any number
if you add then you get zero!The same thing happens a lot; for matrix multiplication, the inverse matrix would be the operation inverse.
For rational numbers (not including zero, of course), the reciprocal would be the multiplicative inverse.
Subsection 8.3.2 What is a group?
Definition 8.3.3. Group.
If a set and binary operation on that set is closed and associative with identity and inverses for every element, we call that set a group.
Example 8.3.4.
The most excellent examples of this are the following:
under addition with zero as identityThe sets
and except zero (written as and respectively) under multiplication with 1 as identity under addition with as identity. For example, in every element has an inverse; and because
Remark 8.3.5.
If we are talking about any old group, we just call it
Also, after a while, it gets boring to always type
Example 8.3.6. A preview of what's to come.
We noted that
Indeed there is, and we will see it soon. But notice that things will be more complicated.
For instance, in
both and have multiplicative inverses (in fact, themselves), so is a (multiplicative) group, just likeBut in
both and do not have multiplicative inverses, so it would not make sense to say that is a group.
That extra complication is one reason we need to think hard about these things!
Subsection 8.3.3 Properties of groups we will need
Subsubsection 8.3.3.1 Solutions to equations
Since a group has inverses, we can solve very simple ‘linear’ equations in them. This is stated asSubsubsection 8.3.3.2 Inverses of product
We can give a formula of sorts for the inverse in any group; see Exercise 8.4.8.Fact 8.3.7.
The inverse of
Proof.
First, \(b^{-1}\) and \(a^{-1}\) exist, so \((b^{-1})(a^{-1})\) exists. Next, if \(ab\cdot x= 1\text{,}\) then
we use associativity to simplify
which gives \(x= b^{-1}a^{-1}\text{.}\)
(Keep in mind that in our main example \(ab\cdot x\equiv 1\) is the notion of equality we are using in finding and using these inverses.)
Subsubsection 8.3.3.3 Finite groups
A group can have finitely many or infinitely many elements. Most of our normal ones, such asSubsubsection 8.3.3.4 Order of a group
Definition 8.3.8.
The number of elements of a finite group is called the order of the group.
For any old group
Example 8.3.9.
So if we are talking about
Subsubsection 8.3.3.5 Order of an element
This is a tougher concept. Suppose you have some element, such asDefinition 8.3.10.
For a group element
Example 8.3.11.
For example, in
So while
Subsubsection 8.3.3.6 The connection
Here comes the coolest part, where we connect the two concepts of order. We will definitely use Theorem 8.3.12 in proving various theorems. Take a look at any old elementTheorem 8.3.12. Lagrange's Theorem on Group Order.
The order of any element
Proof.
Examine the above argument. We have a number of subsets of \(G\text{,}\) all of size \(m\text{,}\) which exactly fill out \(G\text{,}\) which has size \(n\text{.}\) This forces that \(m\) divides \(n\) as integers.
Example 8.3.13.
For example, above we saw that
Subsubsection 8.3.3.7 Cyclic groups
There is another, simpler concept to keep in mind.If
has order and there is some element such that has order as well, then it must go through all the possible other elements of before hittingThis element, whose powers run through all
elements of is called a generator of the group.Any group that has a generator (again, an element whose powers hit all elements of the group) is called a cyclic group.
Subsubsection 8.3.3.8 Abelian groups
This won't come up too much, but it is important to note that most of the groups we will encounter in this course have one additional special property. Namely, it doesn't matter what order you do the operation in. (Such an operation is called commutative.)For instance, clearly (in any
) it is true that or really for any elements at all.Not all groups have this property; you may recall that multiplying matrices in two different orders may yield two different answers.
If your group has this property, then it is clear that Fact 8.3.7 reduces to