Chapter 4
Abelian Groups
This chapter reviews the notion of an abelian group.
4.1
Definitions, Basic Properties, and Some Examples
Definition 4.1
An
abelian group
is a set
G
together with a binary operation
?
on
G
such that
1. for all
a, b
∈
G
,
a ? b
=
b ? a
(commutivity property),
2. for all
a, b, c
∈
G
,
a ?
(
b ? c
) = (
a ? b
)
? c
(associativity property),
3. there exists
e
∈
G
(called the
identity element
) such that for all
a
∈
G
,
a ? e
=
a
(identity
property),
4. for all
a
∈
G
there exists
a
0
∈
G
such that
a ? a
0
=
e
(inverse property).
Before looking at examples, let us state some very basic properties of abelian groups that follow
directly from the definition.
Theorem 4.2
Let
G
be an abelian group with operator
?
. Then we have
1. the identity element is unique, i.e., there is only one element
e
∈
G
such that
a ? e
=
a
for
all
a
∈
G
;
2. inverses are unique, i.e., for all
a
∈
G
, there is only one element
a
0
∈
G
such that
a ? a
0
is
the identity.
Proof.
Suppose
e, e
0
are identities. Then since
e
is an identity, by the identity property in the
definition, we have
e
0
? e
=
e
0
.
Similarly, since
e
0
is an identity, we have
e ? e
0
=
e
.
By the
commutivity property, we have
e ? e
0
=
e
0
? e
. Thus,
e
0
=
e
0
? e
=
e,
and so we see that there is only one identity.
Now let
a
∈
G
, and suppose that
a ? a
0
=
e
and
a ? a
00
=
e
. Now,
a ? a
0
=
e
implies
a
00
?
(
a ? a
0
) =
a
00
? e
.
Using the associativity and commutivity properties, the left-hand side can be written
a
0
?
(
a ? a
00
), and by the identity property, the right-hand side can be written
a
00
. Thus, we have
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