Class 9

Definition:  The Cartesian product of sets  is the set of all ordered n-tuples

The Cartesian product is usually denoted by either    or

Theorem 18: Let <G₁,*₁>,  <G₂,*₂>,  … <G_n,*_n> be a groups.  For and define . Then  is a group, the direct product of the groups G_i  under the binary operation *.

If all the G_i  are commutative, we sometimes use additive notation  and say “the direct sum of the groups G_i.

Theorem 19: The group  is isomorphic to  if and only if m and n are relatively prime, that is, if and only if

Corollary: The groupis cyclic and isomorphic toif and only if the numbers m_i, i = 1…n  are such that the gcd of any two of them is 1.

Theorem 20: Let   If a_i is of finite order r_i  in G_i, then the order of  in  is equal to the least common multiple (lcm) of all the r_i.

Theorem 21: The intersection of any collection of subgroups H_i of a group is a subgroup.

Definition:  Let be a group and be a subset of elements of G.  Then the smallest subgroup of G  that containsis the subgroup generated by .  If this subgroup is all of G, then generates G and are generators of G.  If is finite and generates G, then we say that G is finitely generated.

Theorem 22:  If is a group and , then the subgroup H of G generated by  consists of precisely those elements of G that are finite products of integer powers of the a_i, where powers of a fixed a_i may occur several times in the product.

Proof:

Let K = all finite products of integer powers of the a_i.  Clearly and since H is the smallest such subgroup we simply need to show that K is a subgroup in order to show that K = H.

Note that K is closed because the product of a two distinct finite products of integer powers of the a_i will again be a finite product of integer powers of the a_i.

Then note that  so   Finally, for any element k in K, the inverse element is simply found by reversing the order of the a_i and changing the sign of the exponents on the a_i (see example below).  Such elements are still finite products of integer powers of the a_i, so if

Theorem 23 (Fundamental Theorem of Finitely Generated Abelian Groups):  Every finitely generated abelian group G is isomorphic to a direct product of cyclic groups of the form,

 where the p_i are primes, not necessarily distinct.  The number of factors of (the Betti number) is unique) as are the prime powers   The order of the factors can be rearranged.