9.1 Groups 545 Does every integer have an inverse under the operation of multiplication? For multiplication, the product of an integer and its inverse must yield the multiplicative identity element, 1. What is the multiplicative inverse of 2? That is, 2 times what number gives 1? 2 1 2 1 1 2 ⋅ = ⋅ = However, since 1 2 is not an integer, 2 does not have a multiplicative inverse in the set of integers. Therefore, not every integer has a multiplicative inverse. Group Let’s review what we have learned about the mathematical system consisting of the set of integers under the operation of addition. 1. The set of integers is closed under the operation of addition. 2. The set of integers has an identity element under the operation of addition. 3. Each element in the set of integers has an inverse element under the operation of addition. 4. The associative property holds for the set of integers under the operation of addition. The set of integers under the operation of addition is an example of a group . The properties of a group can be summarized as follows. Profiles in Mathematics Niels Abel (1802–1829), Évariste Galois (1811–1832) Untimely ends: Important contributions to the development of group theory were made by two young men who would not live to see their work gain acceptance: Niels Abel, a Norwegian, and Évariste Galois, a Frenchman. Although their work showed brilliance, neither was noticed in his lifetime by the mathematics community. Abel lived in poverty and died of malnutrition and tuberculosis at the age of 26, just 2 days before a letter arrived with the offer of a teaching position. Galois died of complications from a gunshot wound he received in a duel over a love affair. The 20-year-old man had spent the night before summarizing his theories in the hope that eventually someone would find it “profitable to decipher this mess.” After their deaths, both men had important elements of group theory named after them. The Abel Prize is an award granted annually by the King of Norway for outstanding research in mathematics. It is intended to give mathematicians an award equivalent to the Nobel Prize. Properties of a Group Any mathematical system that meets the following four requirements is called a group . 1. The set of elements is closed under the given operation. 2. An identity element exists for the set under the given operation. 3. Every element in the set has an inverse element under the given operation . 4. The set of elements is associative under the given operation. It is often very time consuming to show that the associative property holds for all cases. In many of the examples that follow, we will state that the associative property holds for the given set of elements under the given operation. Commutative Group The commutative property does not need to hold for a mathematical system to be a group. However, if a mathematical system meets the four requirements of a group and is also commutative under the given operation, the mathematical system is a commutative (or abelian ) group. The abelian group is named after Niels Abel (see the Profiles in Mathemat ics in the margin). Definition: Commutative Group A group that satisfies the commutative property is called a commutative group ( or abelian group ). Timely Tip Note that the set of elements need not be commutative for the mathematical system to be a group. Also note that every element in the set must have an inverse for the mathematical system to be a group. Because the commutative property holds for the set of integers under the operation of addition, the set of integers under the operation of addition is not only a group, but it is also a commutative group.
RkJQdWJsaXNoZXIy NjM5ODQ=