Supplementary Exercises for Chapters 9-11 239 13. Let n > 1 be a fixed integer and let G be a group. If the set H = {x E GI lxl = n} together with the identity forms a subgroup of G, prove that it is a normal subgroup of G. In the case where such a subgroup exists, what can be said about n? Give an example of a non-Abelian group that has such a subgroup. Give an example of a group G and a prime n for which the set H together with the identity is not a subgroup. 14. Show that Q/Z has a unique subgroup of order n for each positive integer n. 15. If H and K are normal Abelian subgroups of a group andH K= {e}, prove that HK is Abelian. 16. Let G be a group of odd order. Prove that the mapping xx from on. OHN L. HESS at: t of H in G. a finite ery posi- G to itself is one-to-one. GG if index of 17. Suppose that G is a group of permutations and orb(5)= {1, 5), prove that stab(5) is normal in G. 18. Suppose that G HX K and that N is a normal subgroup of H. on some set. If IGI 60 al in D Prove that N is normal in G. 19. Show that there is no homomorphism from Z3 Z, Z2 onto up if and 20. Show that there is no homomorphism from Ag onto a group of order 2, 4, or 6, but that there is a homomorphism from Ag onto a me very group of order 3. 21. Let H be a normal subgroup of S of order 4. Prove that S/H is iso- morphic to S3. 22. Suppose that is a homomorphism of U(36), Ker 1, 13, 25}, and d(5) 17. Determine all elements that map to 17. s isomor- 23. Let n = 2m, where m is odd. How many elements of order 2 does D/Z(D, ) have? How many elements are in the subgroup (R360/Z(D, )? How do these numbers compare with the number of elements of order 2 in D? 24. Suppose that H is a normal subgroup of a group G of odd order and that IHI 5. Show that H C Z(G). m 25. Let G be an Abelian group and let n be a positive integer. Let G lg I g" = el and G" to G". laced by s prime? {g" lg E G). Prove that G/G, is isomorphic ww. T 26. Let R+ denote the multiplicative group of positive reals and let T = {a+ biE Cla2 +b2 = 1) be the multiplicative group of complex numbers of norm 1. Show that C is the internal direct product of R and T ich every groups of