Abstract Algebra Theory and Applications - Computer Science ...

46 CHAPTER 2 GROUPS1. mg + ng = (m + n)g for all m, n ∈ Z;2. m(ng) = (mn)g for all m, n ∈ Z;3. m(g + h) = mg + mh for all n ∈ Z.It is important to realize that the last statement can be made only becauseZ and Z n are commutative groups.Historical NoteAlthough the first clear axiomatic definition of a group was not given until thelate 1800s, group-theoretic methods had been employed before this time in thedevelopment of many areas of mathematics, including geometry and the theory ofalgebraic equations.Joseph-Louis Lagrange used group-theoretic methods in a 1770–1771 memoir tostudy methods of solving polynomial equations. Later, Évariste Galois (1811–1832)succeeded in developing the mathematics necessary to determine exactly whichpolynomial equations could be solved in terms of the polynomials’ coefficients.Galois’ primary tool was group theory.The study of geometry was revolutionized in 1872 when Felix Klein proposedthat geometric spaces should be studied by examining those properties that areinvariant under a transformation of the space. Sophus Lie, a contemporary ofKlein, used group theory to study solutions of partial differential equations. One ofthe first modern treatments of group theory appeared in William Burnside’s TheTheory of Groups of Finite Order [1], first published in 1897.2.3 SubgroupsDefinitions and ExamplesSometimes we wish to investigate smaller groups sitting inside a larger group.The set of even integers 2Z = {. . . , −2, 0, 2, 4, . . .} is a group under theoperation of addition. This smaller group sits naturally inside of the groupof integers under addition. We define a subgroup H of a group G to be asubset H of G such that when the group operation of G is restricted to H,H is a group in its own right. Observe that every group G with at least twoelements will always have at least two subgroups, the subgroup consisting ofthe identity element alone and the entire group itself. The subgroup H = {e}of a group G is called the trivial subgroup. A subgroup that is a propersubset of G is called a proper subgroup. In many of the examples that we