Solve

action. 2. Let (F, +, .) be a field. Let G = (F, .) the group. Let V be any vector space over F (Refer chapter on Vector spaces). Define * by a * v = a.v, a e F, v E V, where . is scalar multiplication in V Show that * is a group action. 3. Let S be a non empty set and G be a group of permutations of S, i.e., G = A(S). Show that * defined by 0 * x = 0(x) V x E S, GE G is a group action of G on S 4. (i) Let H be a subgroup of G and = set of all left cosets of H in G. Define * by g * aH = gaH, ge G, aHE . Show that * is a G action. (ii) Let q : G - A() be the homomorphism corresponding to *. Show that Ker q is the largest normal subgroup of G contained in Hand Ker p = n xHx- XEG Hence prove the generalized Cayley's theorem. (See page 156, theorem 18). The set is sometimes denoted by G/H and is called the (left) coset space of G relative to H. (Notice H here is not essentially normal). Again if we wish to work with right cosets, we can define * by g * Ha = Hag 1 (iii) If H o($8) g EG 9, Let G t ting set S. If x