D4 question

**nhk** · April 18th 2010, 02:07 PM

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

**Drexel28** · April 18th 2010, 03:37 PM

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 $C_n$

**nhk** · April 18th 2010, 03:56 PM

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

**Drexel28** · April 18th 2010, 04:09 PM

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?

**nhk** · April 18th 2010, 05:00 PM

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

**Drexel28** · April 18th 2010, 05:54 PM

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 $D_4\cong C_4\ltimes C_2$

**nhk** · April 18th 2010, 06:04 PM

tnahks for your help anyway.

**nhk** · April 18th 2010, 06:05 PM

tonio can you save me again please?

**aliceinwonderland** · April 18th 2010, 06:28 PM

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 $xhx^{-1} = h^{-1}$ .

**nhk** · April 18th 2010, 06:40 PM

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?
Thanks for your help

**aliceinwonderland** · April 18th 2010, 06:53 PM

Originally Posted by nhk

I am not sure why that the internal direct product of abelain subroups makes D4 abelian, could you explain this to me a little?
Thanks for your help

Direct sum - Wikipedia, the free encyclopedia
Direct sum - Wikipedia, the free encyclopedia

**nhk** · April 18th 2010, 09:56 PM

definitions always get me!! THanks for you help aliceinwonderland

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