4. Prove the following assertions.
a) N ? G is a characteristic subgroup iff for all a ? Aut G we have a(N) = N.
b) If N is a characteristic subgroup of G, then N <G.
c) A characteristic subgroup of a characteristic subgroup is a characteristic subgroup.
(cf. with the fact that a normal subgroup of a normal subgroup is not necessarily
a normal subgroup)
d) A characteristic subgroup of a normal subgroup is a normal subgroup.
e) The following subgroups are characteristic in G: Z(G), G', G" = [G', G'], [G, G'].
f) If a Sylow p-subgroup is normal, then it is characteristic.
g) For any n the subgroup generated by all elements of order n is characteristic.
h) In a cyclic group, every subgroup is characteristic.
i) Every subgroup of Q is normal, but not all of them are characteristic.
