Abstract Algebra Theory and Applications - Computer Science ...

48 CHAPTER 2 GROUPSnecessarily obtain another invertible matrix. Observe that( ) ( ) ( )1 0 −1 0 0 0+= ,0 1 0 −1 0 0but the zero matrix is not in GL 2 (R).Example 14. One way of telling whether or not two groups are the sameis by examining their subgroups. Other than the trivial subgroup and thegroup itself, the group Z 4 has a single subgroup consisting of the elements0 and 2. From the group Z 2 , we can form another group of four elementsas follows. As a set this group is Z 2 × Z 2 . We perform the group operationcoordinatewise; that is, (a, b)+(c, d) = (a+c, b+d). Table 2.5 is an additiontable for Z 2 ×Z 2 . Since there are three nontrivial proper subgroups of Z 2 ×Z 2 ,H 1 = {(0, 0), (0, 1)}, H 2 = {(0, 0), (1, 0)}, and H 3 = {(0, 0), (1, 1)}, Z 4 andZ 2 × Z 2 must be different groups.Table 2.5. Addition table for Z 2 × Z 2+ (0, 0) (0, 1) (1, 0) (1, 1)(0, 0) (0, 0) (0, 1) (1, 0) (1, 1)(0, 1) (0, 1) (0, 0) (1, 1) (1, 0)(1, 0) (1, 0) (1, 1) (0, 0) (0, 1)(1, 1) (1, 1) (1, 0) (0, 1) (0, 0)Some Subgroup TheoremsLet us examine some criteria for determining exactly when a subset of agroup is a subgroup.Proposition 2.9 A subset H of G is a subgroup if and only if it satisfiesthe following conditions.1. The identity e of G is in H.2. If h 1 , h 2 ∈ H, then h 1 h 2 ∈ H.3. If h ∈ H, then h −1 ∈ H.Proof. First suppose that H is a subgroup of G. We must show that thethree conditions hold. Since H is a group, it must have an identity e H .We must show that e H = e, where e is the identity of G. We know that