Explain it

l. 2. 12. 13. 14. 15. Show that a group having five or less elements is abelian. For elements a, b, x in a group G. Show that (0 0(a) = 0(0") (a) 0(a) = o(x\"lax). (iii) 0(0") 5 0(a) (iv) 0(ab) = 0(ba) (v) If a e G be the only element of order n then that a 6 2(0), the centre of G. . if a finite group possesses an element of order 2, show that it possesses an odd number of such elements. . Let G be a nite group. Leta e G be such that 0(a) = 0(G). Show that G is cyclic, generated by a. Hence show that a group of order n is cyclic iff it has an element of order n. . Show that every element in Us is its own inverse (so is of order 2) and hence Us is not cyciic. . Let G be a group and a e G. Show that H = = {a"]n an integer} is a subgroup of G and also if K is any subgroup of G s.t., a e K, then H ; K . let a e G be such that 0(a) = mn, where m, n are coprime. Show that a = be, where 0(a) = m, 0(c) = n. (See Problem 48 on page 146). Show that a subgroup (at {8}) of an innite cyclic group is infinite. . Show that elements of nite order in any abelian group form a subgroup. 10. 11. Show that for n > 2, the order of U\" is even. If G is a cyclic group of order p, a prime then show that any non identity element of G is of order 1). Let G be a cyclic group of order 6 generated by a. Let H, K be the subgroups generated by 02, a3 respectively. Prove that 0(H) = 3, 0(K) = 2, G = HK and H n K = {e}. Find order of each element in the group G = {5:1, it} under multiplication. Find all the subgroups of the quaternion group G and show that El no two non-trivial subgroups H, K of G s.t., H n K is identity only. Let A(R) be the group of all permutations on R, where R = set of reals. Letf: R > R s.t.,f(x) = x and g: R ) R s.t., g(x) = I x. Show thatfand g are both elements of order 2