Cyclic Groups Characterizing Groups

respectively. Finally, 0a = 0, where the first 0 is in Z, and the second is in G. It is important to realize that G is still a group with a single group operation, addition, even though we use multiplication in our notation. We shall shift our view later in defining modules and vector spaces. Let us start with two equivalent conditions. Theorem 3.13 Let X be a subset of a nonzero Abelian group G. The following conditions on X are equivalent. a Each nonzero element a in G can be uniquely expressed in the form a = n 1 x 1 + n 2 x 2 + · · · + n r x r for n i = 0 in Z and distinct x i ∈ X. b X generates G, and n 1 x 1 + n 2 x 2 + · · · + n r x r = 0 for n i ∈ Z and x i ∈ X iff n 1 = n 2 = · · · = n r = 0. The conditions should remind the reader of linearly independent vectors. As we will soon find out, this similarity is not accidental. Definition 3.19 free Abelian group An Abelian group having a nonempty generating set X satisfying the conditions in Theorem 3.13 is a free Abelian group and X is a basis for the group. We have already seen a free Abelian group: The finite direct product of the group Z with itself is a free Abelian group with a natural basis. In fact, we may use this group as a prototype. Theorem 3.14 If G is a nonzero free Abelian group with a basis of r elements, then G is isomorphic to Z × Z × · · · × Z for r factors. Furthermore, while we may form different bases for a free Abelian group, all of them will have the same size. Theorem 3.15 rank Let G be a nonzero free Abelian group with a finite ba- sis. Then, every basis of G is finite and all bases have the same number of elements, the rank of G, rank G = log 2 |G2G|. Subgroups of free Abelian groups are simply smaller free Abelian groups. Theorem 3.16 A subgroup K of a free Abelian group G with finite rank n is a free Abelian group of rank s ≤ n. Furthermore, there exists a basis {x 1 , x 2 , . . . , x n } for G and d 1 , d 2 , . . . , d s ∈ Z + , such that {d 1 x 1 , d 2 x 2 , . . . , d s x s } is a basis for K. All subgroups K of a free Abelian group G are normal as it is Abelian. It is clear from Theorem 3.16 that G K is finitely generated: K eliminates gener- ators x i of G when d i = 1 and turns others into generators with finite order d i 1. This statement extends to finitely generated groups, as their subgroups are finitely generated and a similar factorization occurs. The corollary follows. Corollary 3.3 Let G be a finitely generated Abelian group with free part of rank n. Let K be a subgroup of G with free part of rank s ≤ n. Then, GK is finitely generated and its free part has rank n − s. Example 3.7 factoring finitely generated groups Theorem 3.10 factors a finitely generated Abelian group as the product of a free Abelian group and a number of finite cyclic groups. Using Theorem 3.14, we may restate the result of Theorem 3.10: Every finitely generated Abelian group G may be fac- tored into a free Abelian group H and the product of finite cyclic groups T , G = H × T . Then, GT ∼ = H ∼ = Z β , where β is the Betti number of G. T ∼ = T is often called the torsion subgroup of G, and it contains all generators with finite orders.

3.3.2 Rings, Fields, Integral Domains, and Principal Ideal Domains

The concepts of bases and ranks are familiar to most readers from basic linear algebra and vector spaces. There is, indeed, a direct connection, which we will unveil next. We begin by allowing two binary operations for a set. Definition 3.20 ring with unity A ring R, +, · is a set R together with two binary operations + and ·, which we call addition and multiplication, de- fined on R such that the following axioms are satisfied: a R, + is an Abelian group. b Multiplication is associative. c For a , b, c ∈ R, the left distributive law, ab + c = ab + ac, and the right distributive law , a + bc = ac + bc, hold. A ring R with a multiplicative identity 1 such that 1x = x1 = x for all x ∈ R is a ring with unity. Definitions and concepts from groups naturally extend to rings, sometimes with different names. Rather than defining them individually, I list the equiv- alent concepts in Table 3.3. For example, a ring with a commutative multipli- cation operation is called a commutative ring. Using this table, we now define fields, the richest most restrictive structure we will encounter.