Here is the 1st in the series of posts for the Algbera I course I teach now.

An

abelian groupis a pair , where is a set and a binary operation on that is

- associative, i.e. for all ,

and thus we can drop brackets without introducing any ambiguity;- commitative, i.e. for all ,
- there exists , a special element satisfying for all It must not be confused with the integer number 0, as in general it has nothing to do with although it shares similar properties (integers w.r.t. addition are an example of an abelian group, and there these two things are indeed the same). Elements like are called
neutralw.r.t. the operation in question.- for any there exists such that . We denote by . We usually abbreviate as , etc.

Note that these axioms are essentially the axioms of a vector space that “forgot everything about the field”. The following is also very helpful, although might look evident at first sight.

(But it is not! Later, in ring theory, when we see zero-divisors, we will see that the intuition of “cancelling” may fail you!)

The equation for any given has unique solution

Thus we can cancel: iff , for all

**First examples of abelian groups**

- Integers w.r.t. addition as the operation
- Vector spaces w.r.t. addition of vectors as the operation (such a group is called the
*additive group of the vector space*) - additive groups of fields, e.g. the real numbers w.r.t. addition
- Integers modulo w.r.t. addition , i.e.

The size of a group A is usually called *order*, and denote |A|. So while

New abelian groups from old: **direct products,**, a.k.a. **direct sums**

Given two abelian groups and define the group

, thedirect sumof and where the operation is component-wise:

Similarly, one can define direct sums of an arbitarty number of abelian groups. E.g. the additive groups of n-dimensional vector spaces over a field F can be viewed as direct sums of n copies of the additive group of F.

The next important concept is of **homomorphism** and **isomorphism** of abelian groups.

Given two groups and , a function satisfying is called a

homomorphismfrom toWhen is a bijection, then it is called

isomorphism, and one writes , or just

From the point of view of abstract algebra, two isomorphic groups are hardly distinguishable, although of course they can appear in different disguises.

An example of nontrivial isomorphism is

(To see the isomorphism, set and try extending it by additivity, i.e. compute etc. Eventually you arrive at an isomorphism

Interestingly, Can you prove this?)

**Subgroups**

A

subgroupof is a subset of that is closed under the addition in (Notation: or , the latter does not exlcude )

This means that for any also , and immediately implies

Examples of subgroups:

- additive group of a subspace of a vectorspace is a subgroup of in shorthand,
- for any two positive integers one has (this is indeed not immediate, but note that generates a subgroup )
- e.g. for we have that the set of all even integers form a subgroup in the integers.
- for a prime number , define

**Generating sets, etc.**

First, a bit of terminology: any abelian group is a -module, i.e. in a sense relates to in a way similar to the way a vectorspace over a field F relates to F. More precisely, this means that for any , the product is well-defined, as follows:

- if then
- if then
- if then the sum of copies of

Note that, in contrast to vectorspace/field relationship, it can perfectly happen that while neither nor , e.g. for any

Let Define the subgroup generated by to be i.e. the intersection of all the subgroups of containing

One can prove that (Note that in algebra infinite sums are, normally speaking, not allowed, as we don’t have a notion of convergence.)

**Cyclic subgroups.** These are subgroups generated by just one element, i.e. one should take and denote

That is as simple as it gets as far as subgroups are concerned. We define the *order* of to the be the order of the cyclic subgroup it generates.

Similarly, groups generated by one element are called *cyclic*.

One can show the following classification of cyclic groups.

Either (and so ), or for some (and so ).

We postpond a proof until we learned more about homomorphisms.

Every subgroup of a cyclic group is cyclic.

Let and the minimal positive integer such that Suppose that Without loss of generality, as is a subgroup and so By the integer division with a remainder algorithm, one has for On the other hand , as By the choice of we have and so

Generally speaking, complete classification of abelian groups is out of question, but groups that are somehow similar to finite-dimensional vector spaces, *finitely generated* abelian groups, can be completely classified.

An abelian group is called

finitely generatedif there exists such that and

You might like think which of the following groups are finitely generated:

- the direct sum of a finite number of cyclic groups
- for a prime

Finally, note that there are abelian groups that do not admit a countable generating set, leave alone finite. (Hint: to see this, prove first that a group with a countable generating set is countable.)

30 January, 2009 at 12:56 |

[…] of homomorphisms of abelian groups Let be a homomorphism of abelian groups and (we denoted operations in both groups by the same symbol – these are […]

31 January, 2009 at 3:51 |

is Zm a subgroup of Zmn? aren’t they have different addition operator?

31 January, 2009 at 4:44 |

paradox’s comment is valid, to an extent; generates a subgroup