Proving that $D_{12}\cong S_3 \times C_2$

Question

Prove that $D_{12}\cong S_3 \times C_2$.

I really dont know how I should start this question. My gut feeling says in some way I have to consider normal subgroups of $D_{12}$ but I cannot see how this will lead necessarily to a unique solutions.

No full solutions please hints only (partly because I cannot give any more of an attempt and I dont want this downvoted)

Ideally I would like a non geometric solution so that similar techniques can be used for general groups

Answer 1

Show that the subgroup generated by $g^3$ is normal; it's obviously isomorphic to $C_2$. Now show that $ D_{12} / C_2 \cong S_3 $.

(Hint: what does the presentation of $D_{12} / C_2$ look like?)

Then look at the subgroup $H$ of $D_{12}$ generated by $\{g^2,h\}$. You can check $H$ has index $2$, and so is normal, and consequently $D_{12}/H \cong C_2$. Then find the structure of $H$.

Or to put it another way, look at the two homomorphisms, one sending $g^3 \mapsto e$, the other sending $g^2,h \mapsto e$.

Answer 2

You know that $D_{12}$ is the group of symmetries of a Hexagon. Draw the three longest diagonals of the hexagon, and label them $a$, $b$, and $c$. Using this, can you describe a function from $D_{12}$ to $S_3$? What is the kernel of this function? Can you put a copy of $S_3$ back in $D_{12}$?

Answer 3

The usual well known presentation of dihedral groups gives us in this case:

$$D=\left\{\;s,\,t\;:\;\;s^2=t^6=1\,,\,\,sts=t^5 (=t^{-1})\right\}$$

Now, take

$$\;s:=((12)\,,\,1)\;,\;\;t:=((123)\,,\,c)\in S_3\times C_2\;,\;\;C_2=\{1,c\}\;,\;\;c^2=1$$

Observe that $\;s^2=t^6=1\;$ , and

$$sts=(\,(12)(123)(12)\,,\,1c1\,)=(\,(132),\,\,c\,)$$

and

$$((123)\,,\,c)((132)\,,\,c)=((1)\,,\,1)=1\in S_3\times C_2$$

thus....