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 ZWe write [M] in the form Σ z (i) mi, where the z (i) are integers, only finitely many of these numbers are different from zero, and mi passes through the elements of the set M. Such an element is called a "formal sum".

Two such formal sums Σ z (i) mi and Σ z (i) 'mi 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

Σ z (i) mi + Σ z '(i) mi = Σ (z (i) + z '(i)) mi
Are defined.

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) mi) = Σ z (i) α (mi).

If M is a set with exactly t elements, then the free Abelian group with base M is isomorphic to Zt. An isomorphism is obtained Z[M] → Ztby arranging the elements of M, say M = {m1, ..., mt}, and now the element Σ z (i) mi maps onto the t-tuple (z (1), ..., z (t)). Under this figure, the basic element corresponds to mi 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 Figf(M, Z) the (Abelian) group of all mappings M → Z with finite support (with pointwise addition). We can say M as a subset of Figf (M, Z), and we identify every m in M ​​with the function χm with χm(m) = 1 and χm(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

    Σ a (m) m + Σ b (m) m = Σ (a + b) (m) m
    Are defined.

    In this way a "free Abelian group with base M" is obtained.

    Post Comment: The Importance of Free Abelian Groups and Their Bases

    Free Abelian groups are important for two (very different) reasons:
    • 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 Cn(X) = Z[Top (Δn, X)] the singular n-chains in a topological space X. Here it is the set M = Top (Δn, 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 ringZ [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 Z1(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!
    So while it is with free groups that occur naturally, in general no The artificially introduced free Abelian groups have an excellent basis Z[My excellent base: just the set M from which one starts.