Paper 3, Section II, 6D6 \mathrm{D}

Algebra and Geometry
Part IA, 2007

What does it mean to say that a subgroup HH of a group GG 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 HH is a normal subgroup of GG, explain carefully how to make the set of (left) cosets of HH into a group.

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

(i) If GG is cyclic then HH and G/HG / H are cyclic.

(ii) If HH and G/HG / H are cyclic then GG is cyclic.

(iii) If GG is abelian then HH and G/HG / H are abelian.

(iv) If HH and G/HG / H are abelian then GG is abelian.