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What's abelian and purple? An abelian grape!

**Definition 5.6.1**
Let

and

be two elements
of a group

. We say
that

**commutes** with

(or that

** commute**) if

. We call a group

**commutative**
(or

**abelian**) if every
pair of elements

belonging to

commute.
If

is a group which is not necessarily commutative then
we call

**noncommutative**
(or

**nonabelian**).

**Example 5.6.2**
The integers, with ordinary addition as the group
operation, is an abelian group.

Now the reader should understand the punchline to the
joke quoted at the beginning!
**Convention**: When dealing with groups in general we often drop
the and denote multiplication simply by juxtaposition (that
is, sometimes we write in place of and
in place of ( times)). Also, by convention,
, the identity element.
However, if the group is abelian then one often
replaces by , by ,
and then is *not* dropped.
Some typical examples of finite abelian groups:
- the additive group
(= ),
- the multiplicative group
,

where is an integer.
We have
where is Euler's -function.

**Definition 5.6.3**
If

is a multiplicative
group

with only one generator
(i.e., there is a

such that

)
then we say that

is

**cyclic**.

**Lemma 5.6.4**
If

is a finite cyclic group
and

is the smallest integer such that

then

.

**proof**: Since
for any
, we can list all
the elements of as follows:
There are elements in this list.
We have seen (multiplicative) cyclic groups before
in our discussion of the discrete log problem.
The abstract form of the **discrete log problem**
is the following:
Given a finite cyclic group
with generator and given
, find
such that
.

**Exercise 5.6.5**
Show that any group having exactly 2 elements is abelian.

**Exercise 5.6.6**
Write down all the elements of

and compute its multiplication table.

**Exercise 5.6.7**
Write down all the elements of

and compute it's addition table.

**Exercise 5.6.8**
Let

be the subset of all

such that

.
Show that

is non-abelian.

**Exercise 5.6.9**
Show that if

is a group such that

for all

then

is abelian.

** Next:** Permutation groups
** Up:** An introduction to groups
** Previous:** Application: The automorphism group
** Contents**
** Index**
David Joyner
2002-08-23