Paper 3, Section II, $6 \mathrm{D}$

What does it mean to say that a subgroup $H$ of a group $G$ is normal? Give, with justification, an example of a subgroup of a group that is normal, and also an example of a subgroup of a group that is not normal.

If $H$ is a normal subgroup of $G$, explain carefully how to make the set of (left) cosets of $H$ into a group.

Let $H$ be a normal subgroup of a finite group $G$. Which of the following are always true, and which can be false? Give proofs or counterexamples as appropriate.

(i) If $G$ is cyclic then $H$ and $G / H$ are cyclic.

(ii) If $H$ and $G / H$ are cyclic then $G$ is cyclic.

(iii) If $G$ is abelian then $H$ and $G / H$ are abelian.

(iv) If $H$ and $G / H$ are abelian then $G$ is abelian.

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