Finding the order of the set of inner autos of D_4

[Mr Davis 97]

Homework Statement

Find the order of $Inn (D_4)$, where $D_4$ is the set of symmetries of the square.

Homework Equations

The Attempt at a Solution

Is the only way to this by brute force calculation of all of the inner automorphisms, and to see which are distinct?

 

[fresh_42]

You've just proven $Inn(D_4) \cong D_4/Z(D_4)$. So what is the center of $D_4$?

 

[Mr Davis 97]

You've just proven $Inn(D_4) \cong D_4/Z(D_4)$. So what is the center of $D_4$?

I think that the center is $\{R_0, R_{180} \}$. So the order of $D_4 / Z(D_4)$ is 4, which makes the order of $Inn( D_4)$ 4?

 

[fresh_42]

I think that the center is $\{R_0, R_{180} \}$. So the order of $D_4 / Z(D_4)$ is 4, which makes the order of $Inn( D_4)$ 4?

Would have been my guess, too, but this is no proof. And even if we know the order were four, which one of the two groups of order four is it? do you know a representation of $D_4$ as a semi-direct product or which are the normal subgroups of $D_4$?