# Is N + an Abelian group

### Free Abelian groups

Let M be a set. With**Z**[M] we denote them

**free abelian group**with base M. The elements of

**Z**We write [M] in the form Σ z (i) m

_{i}, where the z (i) are integers, only finitely many of these numbers are different from zero, and m

_{i}passes through the elements of the set M. Such an element is called a

**"formal sum"**.

Two such formal sums Σ z (i) m_{i} and Σ z (i) 'm_{i} are exactly the same if the corresponding "coefficients" are always the same, i.e. if z (i) = z (i) 'holds for all i. The addition of two such formal sums is through

_{i}+ Σ z '(i) m

_{i}= Σ (z (i) + z '(i)) m

_{i}

These free Abelian groups have the following **universal property**: If G is any Abelian group and if α: M → G is a set map, then α has a continuation to a group homomorphism α ':**Z**[M] → G, and this continuation is uniquely determined (where α 'is called a continuation of α, if α' (m) = α (m) for all m in M.): One simply sets α '(Σ z (i) m_{i}) = Σ z (i) α (m_{i}).

If M is a set with exactly t elements, then the free Abelian group with base M is isomorphic to **Z**^{t}. An isomorphism is obtained **Z**[M] → **Z**^{t}by arranging the elements of M, say M = {m_{1}, ..., m_{t}}, and now the element Σ z (i) m_{i} maps onto the t-tuple (z (1), ..., z (t)). Under this figure, the basic element corresponds to m_{i} the t-tuple e (i) = (0, ..., 0,1,0, ... 0) (the 1 is in the i-th position).

One uses the term **free abelsche** Group not just for the groups that are in the form **Z**[M] are given, but quite generally for any group G, belonging to a group of the form **Z**[M] is isomorphic. (Let M be a suitable set.)

If you are unsure what such "formal sums" are, you can proceed as follows: As carrier a figure a: M → Z we denote the set of all m in M for which a (m) is not zero. If M is a set, then let Fig_{f}(M, Z) the (Abelian) group of all mappings M → Z with finite support (with pointwise addition). We can say M as a subset of Fig_{f} (M, Z), and we identify every m in M with the function χ_{m} with χ_{m}(m) = 1 and χ_{m}(m ') = 0 for all other m'. Instead of a one then also writes Σ a (m) m and calls this a "formal sum". Since we only consider functions a with finite support, these are formal sums finite sums (= almost all coefficients are zero). Two such formal sums Σ a (m) m and Σ b (m) m are equal if and only if the corresponding "coefficients" of a and b are equal, i.e. if a (m) = b (m) for all m in M holds. The addition of two such formal sums is through In this way a "free Abelian group with base M" is obtained. |

- If one can show from a naturally occurring group G that G is a free Abelian group, then one is happy: because it shows that G has a very clear structure that one can calculate in G without any problems.

Example: If V is a k-vector space, where k is a field of characteristic zero, then every subgroup of V that is generated by a finite number of vectors is a free Abelian group. (When we talk about subgroups of V, we only consider the additive structure of V: every vector space is an Abelian group with regard to addition.)

If one knows that G is a free Abelian group, one will choose a basis and express all elements as linear combinations of elements of this basis. However, it should be emphasized that there will usually be no excellent basis. So you look for a suitable base in which the elements you are interested in have a particularly simple form ... - But there are also cases in which it is the base M that one is interested in and not so much (or actually not at all) the free Abelian group
**Z**[M] itself: The set M is given, and one leads**Z**[M] only in order to be able to calculate with the elements in M, to be able to assign multiplicities to them, ...

Typical examples of this approach:- The group C
_{n}(X) =**Z**[Top (Δ^{n}, X)] the**singular n-chains**in a topological space X. Here it is the set M = Top (Δ^{n}, X) you are interested in. The formal sums are only formed in order to be able to calculate. - If G is a (multiplicatively written) group, then in algebra the associated
**integer group ring**Z [G] considered. (The free Abelian group**Z**[G] becomes a ring by using the multiplication of G!) The formation of**Z**[G] allows group elements of G not only to be multiplied, as can be done in group G (the product is again an element of G.), but also to be added (the sum, however, is not again an element of G, but only a formal sum of such elements ...)

Sometimes the construction of

**Z**[M] just a first step to find a suitable (computational) object for the elements in M.- It is possible that you are only on special elements in M (or in
**Z**[M]) interested, for example in a subgroup U of**Z**[M] (typical example: the formation of Z_{1}(M) in homology theory). - Or you would like to calculate in
**Z**[M] Zero certain elements that one considers superfluous: One then looks instead**Z**[M] a corresponding group of factors**Z**[M] / U ', where U' is a subgroup (the subgroup created by the "superfluous" elements). - Or finally - you do both: you consider a pair of subgroups U, U 'of
**Z**[M], where U 'is contained in U, and the factor group U / U' is formed. This is exactly what is done when introducing homology or cohomology groups!

- The group C

## Post Comment: The Importance of Free Abelian Groups and Their BasesFree Abelian groups are important for two (very different) reasons:no The artificially introduced free Abelian groups have an excellent basis Z[My excellent base: just the set M from which one starts. |

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