(This post is part of the algebra notes series.)
Introduction to groups
These notes are based on the book Contemporary Abstract Algebra 7th ed.
Binary operations
Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G.
In other words, a binary operation on G is a function f
such that f(a,b)=c
with a,b,c as elements of G.
Examples
Addition, subtraction, and multiplication are all binary operations on the set
of integers. These three operations are in fact functions! You can define a
function +
from the set of integers to the set of integers defined by
+(a,b)=the addition of a and b
We can then use infix notation instead of the function notation, and we arrive
with a+b
.
Note that division is not a binary operation on the set of integers. Why
is that? Consider 1
divided by 3
. 1
and 3
are both integers, but 1/3
is not.
Group
Let G be a set together with a binary operation * (meaning a*b is an element of G for all a,b in G). G is called a group under * if the following three properties are all met:
Closure
If * is a binary operation on G, we say G is "closed" under *, or G has "closure" under *
Examples
Prove the integers under addition is a group.
Since the definition of a group explicitly says that groups deal with binary operations, we must first verify that the operation is actually closed.
In this case, there wasn't much to show since the addition of integers is closed and associative under the integers by definition. However, for more complex functions, the proof is not always so easy.
Elementary Properties of Groups
There are some properties that all groups share, no matter what form their elements are in or how complex/simple the operation on that group is.
Thm 2.1: Uniqueness of the Identity
In a group G, there is only one identity element
Proof strategy: Showing uniqueness
A very common strategy in proving the uniqueness of some value is to suppose to the contrary that there are actually two values. You then maneuver your way through to show that the two values you chose are actually equal, meaning there really is just one unique value. In this proof, we will suppose there are two identities, show that they were equal to one another, which lets us conclude that there really is only one identity.
So now we can safely refer to an identify of a group as the identity.
Thm 2.2: Cancellation
In a group G, the right and left cancellation laws hold; ba=ca implies b=c, and abc=ac implies b=c.
We have to take special note to multiply equations on the same side because
groups do not necessarily have the property of being commutative under their
operation. That means there is no guarantee ab
= ba
, so in order to
preserve equality in our arithmetic, we must multiply on the same sides of
both sides of the equation.
Thm 2.3: Uniqueness of Inverses
For each element a in a group G, there is a unique element b in G such that ab=ba=e.
In our initial definition, we stated that there must exist an element b
in
G such that ab
= e
. However, now we're going to show that this inverse is
unique.
Reminder: Proving uniqueness
Recall that a common proof strategy to showing uniqueness of a value is suppose that there are two, and conclude they are the same. In this case, we're trying to show that there is only one inverse of a. Thus, we start out by supposing a has two inverses, and then we work to show they are in fact the same.
Some notation
Now that we have proven the uniqueness of inverses, we can now safely denote the inverse of the element a in the group G by a-1.
When n
is a positive integer, an represents n factors of
aa...a
. Non integer exponents are not defined for this group notation.
(a.5 doesn't mean anything and should thus be avoided).
When n
is negative, gn can be expressed as
(g-1)|n|. For example, g-3 =
(g-1)3.
g0 is defined as the identity e
.
Some familiar laws of exponents hold for groups, such as
- gmn = gm+n
- (gm)n = gmn
Not all laws hold, though. For example, it's not guaranteed that (ab)n equals anbn.
Groups operations are usually represented with a multiplicative notation. Make sure to not get confused when dealing with additive operations!
Thm 2.4: Socks-shoes property
For group elements a and b, (ab)-1=b-1a-1
Proof strategy: Showing equality of elements within groups
Sometimes you want to show that elements a
and b
of a group G are equal.
In the event you know the values, it's pretty straight forward to just compare
them. However, in this case, we know very little about a
and b
, but we can
prove their equality using our knowledge of groups. We recently proved that
the inverse of an element g
in G is unique. This means if ga=e
and gb=e
,
then a
=b
. We use this fact to prove theorem 2.4.