Show that the dihedral group $D_{16}$ is the internal direct product of its Sylow subgroups.

Question

Show that the dihedral group $D_{16}$ is the internal direct product of its Sylow subgroups. (We use the notation $D_{16}$ for the dihedral group of order 32)

Here's what I think.

Since $D_{16}$ is a p-group, then $2^5$ is the order of the Sylow 2-subgroup of $D_{16}$. Which implies that $D_{16}$ is the unique Sylow 2-subgroup of of $D_{16}$. So we're done.

Is that it? It feels like I'm missing something here because I don't think my prof. will gave us this very short/trivial problem. Do you have any ideas?

Answer 1

I agree with you, and I'm similarly mystified by the triviality.

Answer 2

It is definitely as easy as you say. I think his point is that any group is trivially the length one direct product of itself. Easy once you see it, but not "obvious" if you think a direct product must have more than one factor.