# The products of two rational numbers is a rational number.docx

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1. The products of two rational numbers is a rational number.2. Integers are a subgroup of rationals, thus associativity applies as seen inthe exampleabove.3. The multiplicative identitiy for rationals is 1.4. The multiplicative inverse for any rational , where.5. Rational multiplication is commutative. Eg. 2 3 = 3 2.Definition 2.40. A subset S of a group G, is said to be a subgroup of G, if itis agroup itself.Example 2.41. Consider the set of real numbers R, which is a group underaddition. Then the integers Z, which are a subset of real numbers, and alsoform a group under addition (see Example 2.38), are said to a be a subgroupof R.Definition 2.42. Let H be a subgroup of G, and let x G be an element of G,then we define x H, as the subset {x h | h H}, to be a left coset of H, andH x, as the subset {h x | h H} to be the right coset of H. Where is abinary operation, such as addition or multiplication, depending on thedefinition of the group G.
Example 2.43. Consider the set of all even integers 2Z, it is clear that this isa group under addition. We also know that the set of all integers