0.1 Free Abelian Groups22M:201Fall 03J. SimonFREE ABELIAN GROUPS, DIRECT PRODUCTS, FREE GROUPS,FREE PRODUCTSIn this handout, we review some topics in groups that we will use later inthe course. The material on free groups, free products, and presentations ofgroups in terms of generators and relations (see earlier handout on Describinga Group) will be used in the next chapters on computing the fundamentalgroup of the union of two spaces, and on covering spaces. The materialon free abelian groups and direct products will be used constantly in thechapters on homology. This handout is sometimes informal and only coverssome of the material you may need later; so you also should read ChapterIII in the text.0.1 Free Abelian GroupsLet S = {a1, a2, ...}. We want to define the free abelian group with basis S.[This convenient notation for set S gives the illusion that S is countable -that is not intended - S can have any cardinality.] The group G(a1, a2, ...) isto be abelian, contain S, and have no other restricting properties.First adjoin symbols that will represent the identity in G and inversesfor the generating elements: 0, −a1, −a2, .... Now form the set of all abstractfinite sums, e.g. a1+ a2+ a5+ (−a2). Finally, introduce relations (i.e. definethe set of equivalence classes of sums) saying that these elements and sumsbehave like a commutative group; for examplea1+ a2+ a5+ (−a2) = a1+ a5.Since the group is abelian, we can collect like terms; so each element ofthe free abelian group G(a1, a2, ...) can be expressed uniquely as a sumn1a1+ n2a2+ ...where each ni∈ Z and all but at most finitely many niare 0.The group G(a1, a2, ...) is characterized by the following “universal map-ping property”:c2003, J. Simon, all rights reserved page 10.2 Direct ProductsTheorem 1. If A is any abelian group, and f : S → A is any function, thenthere exists a unique extension of f to a homomorphism F : G(S) → A.An algebra digression: Why do we use only finite sums instead of allowinginfinite sums? We certainly can make an abelian group G∞(S) that includesthe finite sums and the infinite sums. However this group will (a) not be gen-erated by S [easy to prove], and (b) not have the universal mapping property[hard to prove]. In particular, the identity function from S into G(S) doesnot extend to a homomorphism of G∞(S) → G(S).w If such a homomor-phism existed, then, G(S) would be a retract [in the group sense, not thetopological sense] of G∞(S) and so [by one of your homework problems, sincethe groups are abelian. . .] we would have G∞(S) expressed as the directsum of G(S) and some other subgroup, none of which is possible.We call the set S a basis for G(S). Any other subset S0of G(S) with theproperty that each element of G(S) can be uniquely expressed as a linearcombination (with coefficients in Z) of elements of S0is also called a basisfor the group.Free abelian groups and their bases are analogous to vector spaces andtheir bases: any two bases have the same cardinality (called the rank ordimension); any bijection between bases extends uniquely to an isomorphism;any function from a basis of a free abelian group[resp. vector space] to anabelian group [resp. vector space] extends uniquely to a homomorphism[resp. linear mapping. There is one subtle difference: in a vector space, anyminimal generating set is a basis; however, this is not true for free abeliangroups. For example, the additive group Z is a free abelian group of rankone, generated by the single element 1; the elements {2,3} also are a minimalgenerating set for Z, but they are not a basis (why?).0.2 Direct ProductsLet S = {G1, G2, ...} be a set of groups (as before, not necessarily countable).The direct productYi=1,2,...Giis the most general group containing all the Gias subgroups that commutewith each other. We can construct the direct product by taking all tuples(g1, g2, ...) , gi∈ Gi,c2003, J. Simon, all rights reserved page 20.2 Direct Productssuch that all but at most finitely many gi= identity ∈ Gi. Define the groupoperation componentwise.When each of the factor groups is isomorphic to the infinite cyclic groupZ, the direct product is just the free abelian group of rank = however manyfactors we have.When there are just a few factor groups, we may denote the product asG1× G2, or sometimes as a sum, G1⊕ G2⊕ G3.Some people use the multiplication notation for strong products (see be-low) and the direct sum notation for weak products. Some reserve the directsum notation for when the summand groups are themselves abelian. Thiscan get confusing, so just make sure to define your notation when you writeyour own papers and books!As with free abelian groups, direct products satisfy a universal mappingtheorem: Given any abelian group W and any list of homomorphisms fi:Gi→ W , there exists a unique extension of all the fito a homomorphism ofthe product of the Giinto W .In defining the direct product of infinitely many groups, we could changethe definition to allow tuples with infinitely many nontrivial terms. This iscalled the strong direct product, in contrast to the one defined above, whichis called the the weak direct product. The strong direct product is an abeliangroup containing all the Gi, but it is not generated by the Giand does notsatisfy the universal mapping condition.Note on notation: We use the symbol × for “direct product” and thesymbol ⊕ for direct sum. But the symbol ⊗ means “tensor product”, whichis very different from direct product.Recall that the “fundamental theorem of finitely generated abelian groups”says each finitely generated abelian group A is the direct product of a finiteabelian group and a free abelian group. The rank of the free part, which isan invariant of the given group A, is called the rank of A. The finite part ofA can itself be expressed as the direct product of cyclic groups. There aremany ways to do this; however, it can be done in a special way that givesmore invariants of A: a finite abelian group can be expressed uniquely as adirect product of cyclic groups where the order of each one divides the orderof the next. This list of orders is an invariant of A.c2003, J. Simon, all rights reserved page 30.3 Free Groups0.3 Free GroupsOnce again, we start with a set S = {a1, a2, ...} and seek to construct a groupF (S) containing S. But this group will not be abelian - it simply will be themost general group possible that contains the set S. As with the