
Normal Subgroup
1. Find all the normal subgroups of <a> of S3 and <b> of D4, where a is in S3 and b is in D4. I know A4 is a normal subgroup of S3. Are there any others?
2. Let H be a subgroup of an arbitrary group G. Prove H is normal iff it has the following property: For all a,b, in G, ab is in H iff ba is in H.

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We can think of $\displaystyle S_3$ as $\displaystyle \{e,a,a^2.b,ab,a^2b\}$ with the relation $\displaystyle ba=a^2b$.
(Here $\displaystyle e=(1),a=(1,2,3),b=(1,2)$).
And here is the subgroup diagram.

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We can think of $\displaystyle D_4$ as $\displaystyle \{e,a,a^2,a^3,b,ab,a^2b,a^3b\}$ with the relation $\displaystyle ba=a^3b$. (Here $\displaystyle e=(1),a=(1,2,3,4),b=(2,3)$.
And here is the subgroup (much nicer looking).