1. ## D4 question

Prove that D4 (Dihedral group) cannot be expressed as an internal direct product of two proper subgroups.
I know that the only two possible subgroups would be the subgroups of order 4 and 2. I am thinking since D4 is not commutative I can get a contradicition this way, but I am not sure how to do it. Any help would be welcomed

2. Originally Posted by nhk
Prove that D4 (Dihedral group) cannot be expressed as an internal direct product of two proper subgroups.
I know that the only two possible subgroups would be the subgroups of order 4 and 2. I am thinking since D4 is not commutative I can get a contradicition this way, but I am not sure how to do it. Any help would be welcomed
I thought the dihedral group $D_n$ is always isomorphic to a semidirect product of $C_2$ and $C_n$

3. what is C2 and Cn? I think we have not got that advanced in the class i am taking right now.

4. Originally Posted by nhk
what is C2 and Cn? I think we have not got that advanced in the class i am taking right now.
What are you doing in class right now?

5. factor groups and normal subgroups are what we are doing in my class right now.

6. Originally Posted by nhk
factor groups and normal subgroups are what we are doing in my class right now.
I'm not sure. You should probably let someone better at group theory (e.g. tonio) take a look at this. I'm pretty sure that $D_4\cong C_4\ltimes C_2$

7. tnahks for your help anyway.

8. tonio can you save me again please?

9. Originally Posted by nhk
tonio can you save me again please?
Well, I might be able to help u too

As u said, the possible proper subgroups of D4 are all abelian (groups of order 2, 4). If D4 are the internal direct product of its nontrivial proper subgroups, then D4 becomes abelian. Contradiction.

As Drexel28 said,

$D_4 \cong C_4 \ltimes_\phi C_2$, where $\phi:C_2 \rightarrow \text{Aut}(C_4)$ and the assoicated action is $x \cdot h = h^{-1}$ for all h in $C_4$ and x in $C_2$ such that $xhx^{-1} = h^{-1}$.

10. Originally Posted by aliceinwonderland
Well, I might be able to help u too

As u said, the possible proper subgroups of D4 are all abelian (groups of order 2, 4). If D4 are the internal direct product of its nontrivial proper subgroups, then D4 becomes abelian. Contradiction.

As Drexel28 said,

$D_4 \cong C_4 \ltimes_\phi C_2$, where $\phi:C_2 \rightarrow \text{Aut}(C_4)$ and the assoicated action is $x \cdot h = h^{-1}$ for all h in $C_4$ and x in $C_2$ such that $xhx^{-1} = h^{-1}$.
I am not sure why that the internal direct product of abelain subroups makes D4 abelian, could you explain this to me a little?