Introduction to Groups

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SET
Sets A collection of "things" (objects or
numbers, etc).Each member is called an element
of the set.There should be only one of each
member (all members are unique). Here is
a set of clothing items...
Examples Set of clothes socks, shoes, pants,
... Set of even numbers ..., -4, -2, 0, 2, 4,
... Positive multiples of 3 that are less than
10 3, 6, 9 set of counting numbers less than 5
1,2,3,4
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Operations An operation takes elements of a
set, combines them in some way,and produces
another element. To define an operation, and one
that makes the most sense is mixing. So for
example, red mixed with green makes yellow, and
red mixed with blue makes purple.
But saying "red mixed with blue makes purple" is
long and annoying. If I have to write a lot, I'm
going to want to shorten that up. So I'm going
to let "mixed with" be symbolized by and
"makes" be symbolized by . So "red mixed with
blue makes purple" becomes "red blue purple".
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Binary Operations A binary operation is just
like an operation, except that it takes 2
elements, no more, no less, and combines them
into one.
You already know a few binary operators, even
though you may not know that you know them 2
3 5 4 3 12 4 - 4 0 These all take two
numbers and combine them in different ways to get
one number. Notice the last example, 4 - 4 0.
It still takes two elements, even if they are the
exact same elements. (Also note division is not
included, because it also returns a
remainder) Now above it looks like there are 3
operations. You will learn in a minute that there
are really only two!
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Well Defined One thing about operators is that
they must be well defined. But reverse that. They
must be defined well. Think about applying those
two words, "defined well" to the English
language. If a word is defined well, you know
exactly what I mean when I say it. The word
"angry" is defined pretty well, as you know
exactly what I mean when I say it. But if I
said the word, "date", would you know that I was
talking about a piece of fruit, rather than the
calendar date? OR
5
Now let's apply this! If I give you two numbers
and a well defined operations, you should be able
to tell me exactly what the result will be.
For example, there is only one answer to 5 3.
That is because the operator is well defined.
But there are some things that look like
operators which aren't well defined.
For example, square roots. When we write x2 25,
or rather x v(25) There are two answers to
this question. If you tell me the answer is 5,
I could just say, "Nope, the answer is -5. You're
wrong." Because 55 25 and (-5)(-5) 25.
With well defined operators, there is only one
possible answer.
Now as a final note with operations, many times
we will use to denote an operation. We don't
mean multiplication, although we certainly can
use it for that. But normally, we just mean "some
operation". When we do mean multiplication, it
will be clear.
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Introduction to Groups
A group is a set combined with an operation So
for example, the set of integers with addition.

• A group is a set G, combined with an operation ,
  such that
• The group contains an identity
• The group contains inverses
• The operation is associative
• The group is closed under the operation.

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IDENTITY Use the operation on any element and
the identity, we will get that element back.
There is only one identity element for every
group. The symbol for the identity element is e,
or sometimes 0.
Formal StatementThere exists an e in the set G,
such that a e a and e a a, for all
elements a in G
INVERSES An element of the group, there is
another element of the group such that when we
use the operator on both of them, we get e, the
identity. There is only one identity element for
every group. Each element in the group has a
different inverse. The notation that we use for
inverses is a-1 .
Formal StatementFor all a in G, there exists b
in G, such that a b e and b a e.
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Commutative, Associative and Distributive
Laws COMMUTATIVE The "Commutative Laws" say
we can swap numbers over and still get the same
answer ...
a b    b a
Example
a b    b a
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ASSOCIATIVE The "Associative Laws" say that it
doesn't matter how we group the numbers (i.e.
which we calculate first) ...
(a b) c    a (b c)
Example
(a b) c    a (b c)
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DISTRIBUTIVE The "Distributive Law" is the BEST
one of all, but needs careful attention. This is
what it lets us do
a (b c)    a b    a c
3 lots of (24) is the same as 3 lots of 2 plus 3
lots of 4
So, the 3 can be "distributed" across the 24,
into 32 and 34
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CLOSED under the Operation Imagine you are
closed inside a huge box. When you are on the
inside, you can't get to the outside.
In that same way, once you have two elements
inside the group, no matter what the elements
are, using the operation on them will not get you
outside the group.
If we have two elements in the group, a and b, it
must be the case that ab is also in the group.
This is what we mean by closed. It's called
closed because from inside the group, we can't
get outside of it. And as with the earlier
properties, the same is true with the integers
and addition. If x and y are integers, x y z,
it must be that z is an integer as well.
Formal Statement For all elements a, b in G, ab
is in G
So, if you have a set and an operation, and you
can satisfy every one of those conditions, then
you have a Group.