《Discrete Mathematics II教学-华南理工》Section 14 Factor-Group Computations and Simple Groups.pdfVIP

III.15 Factor-Group Computations 1 Section III.15. Factor-Group Computations and Simple Groups Note. In this section, we try to extract information about a group G by considering properties of the factor group (or “quotient group”) G/N . We also introduce a new class of groups, which have received much attention. Example. Let G be a group with identity e. Consider the normal subgroup N = {e}. Since N has one element, then all cosets of N have one element—in fact, the coset gN = {ge} = {g }. So the factor group G/N is isomorphic to G. Example. For any group G, the factor group G/G is isomorphic to {e}. This is because there is only one coset of G (treating G as a normal subgroup of G) and so G/G is a group with one element. Note. The above two examples are extreme cases of “collapse” of the cosets of G down to elements of G/N . If G is a finite group and N = {e} is a normal subgroup, then G/N is a smaller group than G and so “may have a more simple structure than G” (using quotation marks when referring to the text’s wording). There is a correspondence between the multiplication of cosets in G/N and the multiplication of their representatives in G (as given in Corollary 14.5). III.15 Factor-Group Computations 2 Lemma. If G is a finite group and N is a subgroup of G where |N | = |G|/2, then N is a normal subgroup of G. Proof. Since all cosets of N must be the same size and the cosets partition G, then there are only two cosets of N , namely N and aN where a ∈ G \ N . Now for any g ∈ G, (1) if g ∈ N then gN = Ng = N , and (2) if g ∈ G \ N then gN = Ng since this is the only coset of N other than N itself. So gN = Ng for all g ∈ G, and N is a normal subgroup. Example 15.4. Consider