If a group is non-abelian, then the identity is the only thing that commutes with everything right?
Not necessarily. Plz read this
Center (group theory) - Wikipedia, the free encyclopedia
Group is abelian IFF Z(G)=G
For example, $\displaystyle G = S_4 \times C_2$. The $\displaystyle C_2$ will commute with everything.
Also, every group of order $\displaystyle p^n$ has a non-trivial centre, $\displaystyle p$ prime, for instance in $\displaystyle D_8$ the rotation by 180 degree commutes with everything and is non-trivial.