Finding a substitution of $\mu_0$ to convert from SI to c.g.s in EM laws In c.g.s, we set $$k=\frac 1{4\pi\epsilon_0}=1$$
which gives $$\epsilon_0=\frac 1{4\pi}$$ 
This conversion works, for example, in Gauss' law: 
in SI $$\vec{\nabla}\cdot\vec{E}=\frac\rho{\epsilon_0}$$ and in c.g.s $$\vec{\nabla}\cdot\vec{E}=4\pi\rho$$
Now, we know $c^2=\frac 1{\epsilon_0\mu_0}$, so in order to convert $\mu_0$ (that usually appears in the SI form) into c.g.s, I thought the substitution would be $$\mu_0=\frac {4\pi}{c^2}$$
However, this doesn't yield the correct laws in c.g.s. 
For example, Amper's law in SI is $$\vec{\nabla}\times\vec{B}=\mu_0\vec{j}$$
But in c.g.s it is $$\vec{\nabla}\times\vec{B}=\frac {4\pi}c \vec{j}$$
And not $$\vec{\nabla}\times\vec{B}=\frac {4\pi}{c^2} \vec{j}$$
as I would have expected. So it seems $\mu_0=\frac{4\pi}c$ is the correct substitution. 
How can that be? Where did one $c$ go? What is another assumption I am missing?
 A: You're assuming $c^2\,\mu_0\,\epsilon_0=1$, which is a formula that holds in SI units but not in all unit systems. I find the best "mnemonic" to keep track of all this stuff is the following generalized unit version of Maxwell's equations, detailed in the "Rationalization" section of the Lorentz-Heaviside unit system Wikipedia page:
$$\begin{align}
\nabla \cdot \mathbf{D} &= \rho / \beta, \\
 \quad \nabla \cdot \mathbf{B} &= 0, \\
 \quad \kappa \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}, \\
 \quad \kappa \nabla \times \mathbf{H} &= \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J} / \beta,
\end{align}$$
where $\beta$ and $\kappa$ are in general dimensionful units that define the unit system in question. From this set it is clear we must in general use:
$$c^2\,\mu_0\,\epsilon_0=\kappa^2$$
and Gaussian units have $\kappa=c;\,\beta=\frac{1}{4\pi}$. The reason for your "anomaly" should now be clear from the above equation. Of course, often people define their time and length units to be the same, in which case $c=1$ and the problem doesn't arise.
I empathize: this is one of the things I am really bad at - I trip up on units all the time too. If you're like me you really need to keep a few sharp mnemonics like the above handy so that you can get an overall view of what you're doing rather than blindly plugging into conversion formulas.
