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# Abelian groups

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