Paper 3, Section II, D

Define the quotient group $G / H$, where $H$ is a normal subgroup of a group $G$. You should check that your definition is well-defined. Explain why, for $G$ finite, the greatest order of any element of $G / H$ is at most the greatest order of any element of $G$.

Show that a subgroup $H$ of a group $G$ is normal if and only if there is a homomorphism from $G$ to some group whose kernel is $H$.

A group is called metacyclic if it has a cyclic normal subgroup $H$ such that $G / H$ is cyclic. Show that every dihedral group is metacyclic.

Which groups of order 8 are metacyclic? Is $A_{4}$ metacyclic? For which $n \leqslant 5$ is $S_{n}$ metacyclic?

*Typos? Please submit corrections to this page on GitHub.*