# Inner Automorphisms of the Symmetries of a Square ($D_4$)

I need some help understanding inner automorphisms and how they apply to the symmetries of a square. The problem is:

Let Inn($G$) be the set of inner automorphisms of some group $G$. Now, consider the group of the symmetries of a square, $D_4$. What are the automorphisms of $D_4$? What is |Inn($D_4$)|?

I understand that the definition of an inner automorphism is $f(x)$ s.t. $f(x)$ = $a^{-1}xa$, and $a^{-1}xa$ = $x$ iff $ax$ = $xa$ (commutative). But I'm still very shaky on the definition, and I don't really quite see how to apply this to the symmetries of a square, $D_4$. Thank you for your help.

• There are only $8$ elements of $D_4$ and thus only $8$ possible inner automorphisms. Simply write them all down and see which one repeat. See if you can think of any autmorphisms of $D_4$ which are not listed. – Alex G. Oct 16 '16 at 5:18
• Ah got it! So, the order of $D_4$ (|Inn($D_4$)|) would just be the total # of inner automorphisms, right? – Max Oct 16 '16 at 5:20