Let $D_6=\langle x,y\mid x^6=y^2=e,xy=yx^5\rangle$. Can $D_6$ be written as direct inner product of two non trivial subgroups?

Question

Let $D_6=\langle x,y\mid x^6=y^2=e,xy=yx^5\rangle$. Can $D_6$ be written as direct inner product of two non trivial subgroups?

So, my task is to find normal subgroups $H_1,H_2$ in $D_6$, such that $H_1 H_2=D_6$ and also that $H_1 \cap H_2=\{e\}$. My attempt was to take $H_1=\{e,x^2,x^4\}$ and $H_2=\{e,x,x^3,x^5\}$, because then these are normal subgroups and $H_1\cap H_2=\{e\}$, but $H_1 H_2\neq D_6$. Any idea on what to take? I didn't know how to continue since whenever I take something with $y$ in one of the subgroups, this is not nesecarilly a normal subgroup then.

Answer 1

Your task is to either find such normal subgroups, or prove that no two such subgroups exist. The question isn't "show that $D_6$ can be written as a direct product", the question is whether it can.

Note that (sub)groups of order less than $6$ are abelian, and the (inner) product of two abelian normal groups is abelian, so if it is possible to write $D_6$ as $HK$ with $H$ and $K$ proper nontrivial subgroups with $H\cap K=\{e\}$, then one them has to be of order $6$ and the other has to be of order $2$.

The only normal subgroup of order $2$ is $\langle x^3\rangle$ (prove it), so that should simplify your search.

Answer 2

$$D_6=\langle x^2,y\rangle\times\langle x^3\rangle.$$