What does it mean to say that a subgroup of a group 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 is a normal subgroup of , explain carefully how to make the set of (left) cosets of into a group.
Let be a normal subgroup of a finite group . Which of the following are always true, and which can be false? Give proofs or counterexamples as appropriate.
(i) If is cyclic then and are cyclic.
(ii) If and are cyclic then is cyclic.
(iii) If is abelian then and are abelian.
(iv) If and are abelian then is abelian.