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I am given $D_{12}=\{1,x,x^2,x^3,x^4,x^5,y,xy,x^2y,x^3y,x^4y,x^5y\}$.

I know that the normal subgroups are as follows: $\langle x\rangle, \langle x^2\rangle,\langle x^3\rangle,\langle x^2,y\rangle,\langle x^2,xy\rangle.$

First, I just need some help understanding why these are the normal subgroups. After this, I would appreciate help identifying how to find the elements within these normal subgroups. I know that $\langle x^3\rangle=\{1,x^3\}$. But some help identifying the others, especially the last two would help a bunch.

After I have this information, I am to identify the quotient groups within $D_{12}$ (which is essentially just $G/\langle x\rangle,\dots, G/\langle x^2,xy\rangle$. After I identify these quotient groups, I am to identify which groups the quotient groups are isomorphic to. How would I go about doing this? It can be as simple as using Cayley tables. I am just lost on how to start.

Shaun
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    All normal subgroups can be listed for all dihedral groups. The most obvious one is $C_{6}=\langle x \rangle$, the cyclic group of order $6$. Since it has index $2$, it is normal. The quotient groups have often an order, where it is clear what group it must be. For example, $D_{12}/C_6$ has order $2$, so it must be isomorphic to $C_2$. – Dietrich Burde Sep 16 '21 at 16:48
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    In the notes of Keith Conrad it is shown that the quotients of $D_n/N$ by normal subgroups are again dihedral groups. Then it is even easier. In any case, the notes contain everything you'll need. – Dietrich Burde Sep 16 '21 at 16:54

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