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a) In $D_4$, find $H_1$ isomorphic to $\mathbb{Z}_4$ and $H_2$ isomorphic to $V$, the Klein Four group, with $D_4/H_1$ isomorphic to $D_4/H_2$.

b) In $D_4$, find subgroups $H$ and $K$ with $H$ normal in $K$ and $K$ normal in $D_4$ but $H$ not not normal in $D_4$.

-and in this I'm just confused.

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    Yeah, I just looked at it again and that can't be in D4. I'm just very confused. I'll take that out.2012-12-14

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Some stuff that will help here: $D_4$ has a cyclic subgroup of order $4$ and all other elements have order $2$ (the subgroup of order $4$ is the one given by the rotations).

So we already have a subgroup of order $4$ which is cyclic.

To find one of order $4$ which is not cyclic, we note that the element of order $2$ in the above mentioned cyclic subgroup is central, so if we let $x$ be this element and $y$ be any other element of order $2$, then $\{e,x,y,xy\}$ is a subgroup of order $4$ which is not cyclic.

The part about the quotients by these two subgroups being isomorphic is automatic, as they both have order $2$, and there is only one group of order $2$.

For the second part, we can take the non-cyclic subgroup from above and the subgroup of that generated by $y$. Since $y$ has order $2$ and is not central, the subgroup generated by $y$ cannot be normal.