There are many different areas of mathematics: geometry, number theory,
topology, algebra - and the problems in these fields can be very different from
one another. But in the late 1800s, people noticed they were using the same tool to
solve many different kinds of problems. This sparked an idea: instead of
reinventing the wheel, why not lift the tool out of the different branches of
math and study the full power of the tool in a general form? This led to the
creation of a new subject called Abstract Algebra, and the star of this
field is a structure called the Group.
Before we give the definition of a group, let's see a few examples. This will
highlight the wide variety of groups and help we appreciate why it took a while
for mathematicians to recognize that they were using the same tool in
completely different settings. To begin, imagine you live on an alien planet
where each day is seven hours long. Here, a clock has seven hours. You might be
tempted to number them one through seven, but if computer science has taught us
anything is that you should begin counting with zero. So we'll number the
hours zero through six. As far as the clock is concerned, there are just seven
numbers: zero through six. You can add and subtract any two of these numbers. For
example, what is 3 plus 5? If the hour-hand starts at 3 and you add 5 more
hours, you end up at 1. So on this clock 3 plus 5 equals 1. Similarly, 4 plus 3
equals 0, because if you start at 4 and add 3 more hours you end up at the top,
which is 0. You can subtract as well. To find 1 minus 3, we start at 1 and move
counterclockwise 3 hours. You end up at 5, so 1 minus 3 equals 5.
Clock arithmetic is also called modular arithmetic. In the example we just saw, there are 7 hours, so
we call them the integers mod 7. Mod is short for modular. At first glance, you
might think this is just an amusing number game - but it has many applications
in math and computer science. For our next example, let's look at an
equilateral triangle. We'll number the three corners one, two,
and three. Here is a puzzle: how many different ways can you pick up the
triangle rotate and flip it around and then set it back down so that it's in
the same starting orientation? Let's count! You get the first transformation
by picking up the triangle and setting it back down again unchanged. We'll
call this transformation one. Let's now find the other ways you can rotate and
flip this triangle so that it overlaps itself. For starters, you can pick up the
triangle and rotate it 120 degrees clockwise. This gives you three one two.
We'll call this transformation R, for rotation. If you perform two such
rotations, you get two three one. We call this transformation R squared, since we
applied R twice. And if you do three rotations, you return to the starting
position: one two three. This means that R cubed equals one. Next,
if you pick up the triangle and flip it about the vertical axis you get one
three two. We'll call a flip about the axis F for flip. If you flip it twice, you
return to the starting position, so f squared equals one. You could also flip
then rotate, which gives two one three. And finally, you can flip then perform
two rotations, which gives three two one. This covers all possible transformations.
There are six altogether. Notice that there were two basic transformations: a
120 degree rotation which we called R, and a flip about the vertical axis which
we called F. Aside from the unchanging transformation, each of the
transformations are combinations of these two actions. Notice you can
multiply any two transformations by first applying one, then the other. In
particular, if you multiply any transformation by one, you get the same
transformation. This is because one doesn't actually change the triangle.
This is one way to study a shape in geometry: look at all the ways to
transform the shape onto itself. These transformations are called symmetries.
Believe it or not, there is a close relationship between the symmetries of a
shape and clock arithmetic. Before we see this connection, let's look at one final
example: the integers under addition. The integers, which we denote by the letters
Z with a double stroke, are the set of all whole numbers. Let's
look at a few interesting properties of the integers: You can add and subtract
any two integers, but notice that subtraction is really just adding with
negative numbers. For example, 7 minus 11 is the same as 7 plus negative 11, which
is negative 4. For this reason, it's common to say there's a single operation:
addition. If you add any two integers, you get another integer. We say the integers
are closed under addition. But if you divide one integer by another, you may
not get an integer. For example 3 divided by 5 is 3/5, which is not an integer. So we say
the integers are not closed under division, because when you divide two
integers you may get a number outside the set of integers. And there's one
number which has a unique property: 0. If you add 0 to any integer you get the
same integer. It has no effect when adding. So we've now seen three very
different examples: clock arithmetic, which is more commonly known as modular
arithmetic; the symmetries of a triangle; and the integers under addition. Let's
now see what these three examples all have in common. To begin, each of these
examples have a set of objects which we call elements. For the integers mod 7, the
set of elements are the numbers 0 through 6. For the triangle example, the
set is not the triangle but the 6 geometric transformations, which we also
call symmetries. And for the integers, the set is the infinite collection of whole
numbers. Next, for each of these sets there's a way to combine any two
elements to get another element in the set. We call this an operation, and we use
a symbol to show that we are combining two elements. For the integers, the
operation is addition and the symbol is a plus sign. For the integers mod seven,
the operation is also addition. In modular arithmetic, we also use the plus
sign, even though addition means something different on a clock. As with
the integers, subtraction is redundant. Any subtraction problem can be
re-written as an addition problem. For example, 2 minus 4 is the same as 2 plus
3, and for the symmetries of a triangle, it's common to use a multiplication
symbol. You could use a different symbol if you'd like, but
people tend to reuse the addition and multiplication symbols a lot with groups.
It's also important to remember that each of the three sets are closed under
the operation. That is, if you pick any two elements A and B in the set and
combine them, you get another element in the set. All three structures also have
something called an identity element, this is an element that has no effect
when combined with other elements. For the integers mod seven it's zero. If you
take any number X on the clock and add zero you get X. For the triangle, the
identity element is the transformation one. This is a transformation where you
pick up the triangle and drop it unchanged. If you multiply any
transformation Y by the transformation one you get Y. And for the integers, the
identity element is also zero. For any integer Z, Z plus zero equals Z and 0
plus Z equal Z. There are two other important properties these examples all
share. For every element, there's an opposite element which we call the
inverse. For the integers under addition, the inverse of 3 is negative 3; the
inverse of 7 is negative 7; and so on. And if you add an element and its inverse, you
get 0. The identity element for the symmetries each transformation has an
opposite as well. For example the inverse of the element R that rotates 120
degrees clockwise is the transformation that's a double rotation which is R
squared this is because doing 3 rotations makes a complete circle
returning you to the starting position. In other words, if you multiply R and R
squared you get 1 the identity element. For the symmetries for the integers mod
7 the inverse of 3 is negative 3 but negative 3 is just 4 on the clock so we
say the inverse of 3 is 4 and 3 plus 4 equals 0 the identity element. So for all
three examples, each element X has an inverse.If you're using the plus sign
for your operation, the inverse is written as negative X. And if you're
using the multiplication sign for your operation, the inverse is written as X to
the negative 1. Combining X with its inverse gives you the identity element.
There's one final property that's essential in the definition of a group -
that's the associative property. This just says that when combining three
elements, it doesn't matter how you group them. You can start by combining the
first two elements, or you can start with the last two elements. Either way, you'll
end up with the same answer. This is one of those properties people take for
granted, but it's important nonetheless. If grouping did make a difference, much
of mathematics would come to a screeching halt. Let's now give the
general definition of a group: a group is a set of elements G. It has an operation
which allows you to combine any two elements. Common symbols for the
operation are the plus sign or multiplication sign, but when speaking
generally we'll use an asterisk. Next, the group is closed under this operation.
This means if you combine any two elements in the group, you get another
element in the group. Each element X has an inverse. This is an object that has
the opposite effect of X. And when you combine X and its inverse, you get the
identity element which we'll call e. If you combine any element Y with the
identity element e, you get Y. And finally, the elements obey the associative
property. This is the definition of a group. It took mathematicians many years of
work to identify these as the most essential properties. For example, notice
that groups are not required to be commutative. That is, x times y may not
equal Y times X. This rule was not included because it would exclude many
important examples like the group of symmetries with the triangle. For example,
a flip followed by a rotation gives you something different than a rotation
followed by a flip. If a group is commutative, we call it a commutative
group. Another common name is an abelian group. If the group is not commutative
then we say it's non-commutative or non-abelian.
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