Proving D4 Cannot be Expressed as Internal Direct Product

[tyrannosaurus]

Homework Statement

Prove that D4 (Dihedral group) cannot be expressed as an internal direct product of two proper subgroups.

Homework Equations

The Attempt at a Solution

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

 

[Dick]

A direct sum can be nonabelian. But why not if the order of the factor groups are 2 and 4?

 

[Dickfore]

A direct sum can be nonabelian. But why not if the order of the factor groups are 2 and 4?

The question states direct product, not sum.

 

[Dick]

The question states direct product, not sum.

Is there a difference when you are talking about groups? There is only one binary operation. That's a silly comment.

 

[Dickfore]

Is there a difference when you are talking about groups? There is only one binary operation. That's a silly comment.

What's a direct product of two groups and what's a direct sum?

 

[Dick]

What's a direct product of two groups and what's a direct sum?

The distinction is unimportant. You only choose to say one or the other depending on whether you are using the additive notation for the group operation or the multiplicative. The result is the same. Your comments are not very helpful.

 

[Dickfore]

The distinction is unimportant. You only choose to say one or the other depending on whether you are using the additive notation for the group operation or the multiplicative. The result is the same. Your comments are not very helpful.

Neither are yours.

 

[Dick]

Neither are yours.

Do you think there is a difference? What might it be?