Please Solve This Question

2. Let H be a subgroup of G. Show that T(H) is subgroup of G for all Te Aut G. 3. If H is normal in G, show that T(H) is also normal in G for all TE Aut G. 4. Show that I(G) is a normal subgroup of Aut G. 5. Show that I(G) = { / } if and only if G is abelian. 6. If G Z(G) is cyclic then (show that) G is abelian (See page 111). Deduce that G is abelian if Aut G is cyclic. 7. Show that G = I (G) if G = S3. 8. Show that the commutator subgroup of a group G is a characteristic subgroup of G. 9. If G, = G2, show that I (G,) = I(G,) and Aut G, = Aut G2 10. If G is (non-cyclic) abelian group of order 4, then show that Aut G = S;. (By solved problem 11 and this exercise it follows that two non-isomorphic groups may have isomorphic automorphism groups). 11. If G is a finite group and Te Aut G s.t., T(x) = x if and only if x = e (such an automorphism is called a fixed point free automorphism). Show that g E G can be written as x T (x) for some x E G. 12. Further in exercise 11, if 72 = I, show that G is abelian and o(G) = odd. 13. Let o(G) = 2n. Let H be a subgroup of G consisting of only those elements of G whose order is not 2. Suppose o(H) = n. Show that n is odd and H is abelian. 14. If H is a characteristic subgroup of G, show that T(H) = H for all Te Aut G. 15. (i) If H is a characteristic subgroup of K and K is normal in G, show that H is normal in G. (ii) Show if H is characteristic subgroup of K and K is characteristic subgroup of G then H is characteristic subgroup of G, i.e., the characteristic property is transitive. 16. If K is a characteristic subgroup of G and Te Aut G, show that T' G G K 'K S.t., T'(Kg) = K(Tg) for all ge G is an automorphism. Further, show that the map T - T' is a homomorphism of Aut G into Aut (G/K). 17. Let G be an abelian group. Let H = {x e G|x" = e, n = fixed integer} . Show that H is a characteristic subgroup of G. 18. If H is a unique subgroup of order m in G, show that H is characteristic subgroup of G. 19. Show that every subgroup of a finite cyclic group is characteristic subgroup. 20. Prove that automorphism group of a finite group is finite