TITLE: what are the normal subgroups of the group $S_3 \times S_3$?
QUESTION [5 upvotes]: Find all normal subgroups of $S_3 \times S_3$.

What are normal subgroup and $S_3 \times S_3$? Could I have some examples?

REPLY [5 votes]: Finding some normal subgroups is not hard -- in addition to the ones Arturo mentioned there is (at least) also the group consisting of $(a,b)$ where the permutations $a$ and $b$ have the same sign. However, groping about for examples is not going to help you prove that you have all normal subgroups.
For that, it seems feasible in this case to remember that a normal subgroup is a union of conjugacy classes. Since $S_3$ has only 3 conjugacy classes (the identity, the transpositions, and the 3-cycles), there are 9 conjugacy classes in $S_3\times S_3$, and with a bit of effort you can systematically check which combinations of them are subgroups. You can reduce the number of cases to check significantly by noting that $\{(1,1)\}$ must always be included and that the sum of the sizes of the conjugacy classes you include must divide 36.