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**Definition 3.1.1.**
A **binary operation** * on a set S is a function
* : S × S -> S defined on the set S × S
of all ordered pairs of elements in S and taking values in S.

The operation * is said to be
**associative** if

a * (b * c) = (a * b) * c

for all a,b,c in S.An element e in S is called an

a * e = a and e * a = a

for all a in S.If * has an identity element e, and a is an element of S, then an element b in S is said to be an

a * b = e and b * a = e.

**Proposition 3.1.2.**
Let * be an associative, binary operation on a set S.

**(a)**The operation * has at most one identity element.**(b)**If * has an identity element, then any element of S has at most one inverse.**(c)**If * has an identity element and elements a,b in S have inverses a^{-1}and b^{-1}, respectively, then the inverse of a^{-1}exists and( a

and the inverse of a * b exists and^{-1})^{-1}= a,( a * b )

^{-1}= b^{-1}* a^{-1}.

**Definition 3.1.3.**
A **group** (G,**·**)
is a nonempty set G together with a binary operation
**·**
on G such that the following conditions hold:

**(i)***Closure:*For all a,b in G, the element a**·**b is a uniquely defined element of G.**(ii)***Associativity:*For all a,b,c in G, we havea

**·**(b**·**c) = (a**·**b)**·**c.**(iii)***Identity:*There exists an**identity element**e in G such thate

for all a in G.**·**a = a and a**·**e = a**(iv)***Inverses:*For each a in G there exists an**inverse**element a^{-1}in G such thata

**·**a^{-1}= e and a^{-1}**·**a = e.

We will usually write ab for the product
a **·** b.

**Example** 3.1.1.
The set **Q**^{×} of nonzero rational numbers,
the set **R**^{×} of nonzero real numbers,
and the set **C**^{×} of nonzero complex numbers
form groups under ordinary multiplication.

**Definition 3.1.4.**
The set of all permutations of a set S is denoted by Sym(S).

The set of all permutations of the set {1,2,...,n}
is denoted by S_{n}.

**Proposition 3.1.5.**
If S is any nonempty set,
then Sym(S) is a group under the operation of composition of functions.

**Proposition 3.1.6.**
(Cancellation Property for Groups)
Let G be a group, and let
a,b,c be elements of G.

**(a)**If ab = ac, then b = c.**(b)**If ac = bc, then a = b.

**Proposition 3.1.7.**
If G is a group and a,b belong to G,
then the equations ax = b and xa = b have unique solutions.

Conversely, if G is a nonempty set
with an associative binary operation
in which the equations ax = b and xa = b have solutions
for all a,b in G, then G is a group.

**Definition 3.1.8.**
A group G is said to be
**abelian**
if ab = ba for all elements a,b in G.

**Definition 3.1.9.**
A group G is said to be a
**finite**
group if the set G has a finite number of elements.
In this case, the number of elements is called the
**order**
of G, denoted by |G|.

**Example** 3.1.3.
**Z**_{n}
is an abelian group under addition.

**Example** 3.1.4.
**Z**_{n}^{×}
is an abelian group under multiplication.
Its order is given by the value
(n)
of Euler's phi-function.

**Definition 3.1.10.**
The set of all invertible
n × n
matrices with entries in **R** is called the
**general linear group**
of degree n over the real numbers, and is denoted by
GL_{n}(**R**).

**Proposition 3.1.11.**
The set GL_{n}(**R**)
forms a group under matrix multiplication.

From one point of view, the axioms for a group give us just what we need to work with equations involving the operation in the group. For example, one of the rules you are used to says that you can multiply both sides of an equation by the same value, and the equation will still hold. This still works for the operation in a group, since if x and y are elements of a group G, and x = y, then a ·: x = a · y, for any element a in G. This is a part of the guarantee that comes with the definition of a binary operation. It is important to note that on both sides of the equation, a is multiplied on the left. We could also guarantee that x · a = y · a, but we can't guarantee that a · x = y · a, since the operation in the group may not satisfy the commutative law.

The existence of inverses allows cancellation
(see Proposition 3.1.6 for the precise statement).
Remember that in a group there is no mention of division,
so whenever you are tempted to write a ÷ b or a / b,
you must write a · b^{-1} or b^{-1} · a.
If you are careful about the side on which you multiply,
and don't fall victim to the temptation to divide,
you can be pretty safe in doing the familiar things to an equation
that involves elements of a group.

Understanding and remembering the definitions will give you one level of understanding. The next level comes from knowing some good examples. The third level of understanding comes from using the definitions to prove various facts about groups.

In the study of finite groups,
the most important examples come from groups of matrices.
I should still mention that the original motivation for studying groups
came from studying sets of permutations,
and so the symmetric group S_{n} still has an important role to play.

**22.**
Use the dot product
to define a multiplication on **R**^{3}.
Does this make **R**^{3} into a group?
*Solution*

**23.**
For vectors (a_{1},a_{2},a_{3}) and
(b_{1},b_{2},b_{3}) in **R**^{3},
the cross product is defined by

(a_{1},a_{2},a_{3}) **×**
(b_{1},b_{2},b_{3})
= (a_{2}b_{3}-b_{3}a_{2},
a_{3}b_{1}-a_{1}b_{3},
a_{1}b_{2}-a_{2}b_{1}).

**24.**
On the set G = **Q**^{×} of nonzero rational numbers,
define a new multiplication by

a * b = ab/2, for all a,b in G.

Show that G is a group under this multiplication.
**25.**
Write out the multiplication table for **Z**_{9}^{×}.
*Solution*

**26.**
Write out the multiplication table for **Z**_{15}^{×}.
*Solution*

**27.**
Let G be a group,
and suppose that a and b are any elements of G.
Show that if (ab)^{2} = a^{2} b^{2}, then ba = ab.
*Solution*

**28.**
Let G be a group,
and suppose that a and b are any elements of G.
Show that (aba^{-1})^{n} = a b^{n} a^{-1},
for any positive integer n.
*Solution*

**29.**
In Definition 3.1.3 of the text,
replace condition (iii) with the condition that there
exists e in G such that e · a = a for all a in G,
and replace condition (iv) with the condition that
for each a in G there exists a' in G with a' · a = e.
Prove that these weaker conditions
(given only on the left)
still imply that G is a group.
*Solution*

**30.**
The previous exercise shows that in the definition of a group
it is sufficient to require the existence of a left
identity element
and the existence of left inverses.
Give an example to show that it is *not* sufficient
to require the existence of a left identity element
together with the existence of *right* inverses.
*Solution*

**31.**
Let F be the set of all
*fractional linear transformations*
of the complex plane.
That is, F is the set of all functions

f(z) : **C** -> **C**,
with f(z) = (az+b)/(cz+d),

**32.**
Let G = { x in **R** | x > 1 } be the set of all real numbers
greater than 1.
Define

x * y = xy - x - y + 2, for x, y in G.

(a) Show that the operation * is closed on G.(b) Show that the associative law holds for *.

(c) Show that 2 is the identity element for the operation *.

(d) Show that for element a in G there exists an inverse a

**Lab 1.**
Exercise 3.1.4 in the text asks you
to write out the addition table for Z_{8}.
** Groups15**
also gives a table for Z

**Lab 2.**
Exercise 3.1.5 in the text asks you
to write out the multiplication table for Z_{7}^{×}.
This is an abelian group of order 6,
so you can compare your multiplication table
with the one for Z_{6} produced by
** Groups15**.
Is the pattern the same?
If not, can you rearrange your list of elements
so that the multiplication table

**Lab 3.**
In the list of groups given by
** Groups15**,
find the smallest one that is not abelian.
(This is related to Exercise 3.1.14 in the text,
which asks you to prove that a nonabelian group
must have at least 5 different elements.)
In this group, find an example of two elements x and y for which

(xy)^{2}
x^{2}y^{2}.

(ab)^{n}=a^{n}b^{n}

**Lab 4.**
In the group of order 12 called A_{4} in
** Groups15**,
pair up the elements with their inverses.
Is any element its own inverse?
Find an example of two elements x,y for which

(xy)^{ -1}
x^{ -1}y^{ -1}.

(ab)^{ -1} = a^{ -1}b^{ -1}

**Lab 5.**
Exercise 3.1.21 in the text asks you to prove that if G is a finite group
with an even number of elements,
then there must exist an element a in G (different from the identity e)
for which a^{2} = e.

As experimental evidence pointing to the truth of the exercise,
show the following for each group listed by
** Groups15**:

if the order of the group is an even number, then there is a nontrivial solution to the equation x

Also show that if the order of the group is an odd number, then the only solution to the equation x

Forward to §3.2 | Back to §2.3 | Up | Table of Contents

Forward to §3.2 | Back to §2.3 | Up | Table of Contents