Define the quotient group , where is a normal subgroup of a group . You should check that your definition is well-defined. Explain why, for finite, the greatest order of any element of is at most the greatest order of any element of .
Show that a subgroup of a group is normal if and only if there is a homomorphism from to some group whose kernel is .
A group is called metacyclic if it has a cyclic normal subgroup such that is cyclic. Show that every dihedral group is metacyclic.
Which groups of order 8 are metacyclic? Is metacyclic? For which is metacyclic?