[Math] Finding a normal and not normal subgroup of $S_3$

abstract-algebrafinite-groupsgroup-theory

I'm being asked to find 2 subgroups of $S_3$, one of which is normal and one that isn't normal. I guess, to find the non normal subgroup is easier.

I would do this by trial and error, but since the group is $S_3$ and easily visualizable, I guess that there should be a geometrical property that makes it easy to find a normal subgroup. Any of you guys know one?
Or should I try this by brute force?

Best Answer

HINT: Keep in mind that $|S_3|=6$, so if you have any subgroup of order 3, it has index 2 and is therefore normal. Can you find a subgroup of order 3?

To find a nonnormal subgroup, you will need a different divisor of 6 that is not trivial, and the only choice there is $2$, so you are looking for a 2 element subgroup. Can you find that?

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