Groups
Definition
A group G,  is a set G, closed under a binary operation , such that
the following axioms are satisfied:
1) Associativity of :
For all a, b, cG, we have (a  b)  c = a (b  c).
2) Identity element e for :
There is an element e in G such that for all x G, e  x = x  e = x
3) Inverse a’ of a:
For each a G, there is an element a’ in G such that
a  a’ = a’  a = e.
Examples
Definition
A group G is abelian if its binary operation is commutative.
Example:
• The set Z+ under addition
is not a group. since there is no identity element for + in Z+.
• The set of all nonnegative integers under addition
is not a group since there is no inverse for 2.
• The set Z, Q, R, and C under addition
are abelian groups.
• The set Z+ under multiplication
is not a group since there is no inverse of 3.
Examples
• The sets Q+ and R+, Q*, R*, C* under multiplication
are abelian groups.
Example: Let  be defined on Q+ by a  b=ab/2. Determine Q+ under
such a binary operation  is a group.
It is a group since it satisfies the three properties of a group:
(1) (a  b)  c= (ab/2)  c=(abc)/4, and a  (b  c)= a  (bc/2)=(abc)/4.
Thus  is associative.
(2) a  e=ae/2=a implies e=2. We have 2 a=a 2=a for all a  Q+. So 2
is an identity element for .
(3) a  a’=aa’/2=e=2 implies a’=4/a. We have a  4/a=4/a  a=2. So
a’=4/a is an inverse for a.
Hence Q+ with the operation  is a group.
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Elementary Properties of Groups
Theorem 4.15
If G is a group with binary operation , then the left and right cancellation
laws hold in G.
that is, a b=a c implies b=c, and b a=c a implies b=c for all a, b, c G.
Proof: Suppose a * b = a * c. Then there exists an inverse of a’ to a. Apply
this inverse on the left,
a’ * (a * b) = a’ *(a * c)
By the associatively law,
(a’ * a ) * b = (a’ * a) * c
Since a’ is the inverse of a, a’ * a =e, we have
e*b=e*c
By the definition of e,
b=c
Similarly for the right cancellation.
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Theorem 4.16
If G is a group with binary operation , and if a and b are any elements
of G, then the linear equations a  x=b and y  a=b have unique
solutions x and y in G.
Proof: First we show the existence of at least one solution by just
computing that a’  b is a solution of a  x=b. Note that
a * (a’ * b) = (a *a’) * b, associative law,
=e * b,
definition of a’,
=b,
property of e.
Thus x= a’  b is a solution a  x=b. In a similar fashion, y=b  a’ is a
solution of y  a=b.
To show uniqueness of y, we assume that we have two solutions, y1
and y2, so that y1 a=b and y2 a=b. Then y1 a=y2 a, and by
Theorem 4.15, y1=y2. The uniqueness of x follows similarly. #
Theorem 4.17
In a group G with binary operation , there is only one element e in G
such that
ex=xe=x
for all x G. Likewise for each a G, there is only one element a’ in G
such that
a’  a = a  a’ = e
In summary, the identity element and inverse of each element are
unique in a group.
Proof: We’ve shown the uniqueness of an identity element for any
binary structure in section 3.
Uniqueness of an inverse
Suppose that a G has an inverses a’ and a’’ so that a’ a = a a’ = e
and a’’ a = a a’’ = e. Then
a a’’= a a’ = e
And, by Theorem 4. 15,
a’’=a’
So the inverse of a in a group is unique.
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Corollary
Let G be a group. For all a, b G, we have (a b)’ = b’ a’.
Proof: in a group G, we have
(a b)  (b’ a’) = a (b  b’) a’ = (a e) a’= a  a’=e.By theorem 4.17,
b’ a’ is the unique inverse of a  b. That is, (a b)’ = b’ a’.
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Group Table
Every group table is a Latin square; that is, each element of the group
appears exactly once in each row and each column.
Proof: On the contrary, suppose x appears in a row labeled with a twice.
Say x=a  b and x=a  c. Then cancellation gives b=c. This contradicts
the fact that we use distinct elements to label the columns.
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Finite Groups
One-element Group
{e} with the binary operation  defined by e e=e
Two-element Group
Example: {e, a}, try to find a table for a binary operation  on {e, a} that
gives a group structure on {e, a}.
 e a
e e a
a a x
Since every element can occur exactly once in each row and each
column, we have x =e .
Three-element group
Three-element group
Example: {e, a, b}, try to find a table for a binary operation  on {e, a, b}
that gives a group structure on {e, a, b}.
 e a b
e e a b
a a x y
b b z w
Since every element can occur exactly once in each row and each
column, we have x=b, y=e, z= e, w=a.
Note: There is only one group of three elements, up to isomorphism.