Question

Do permeability and relative permeability have the same dimensions?

Answer 1

Magnetic permeability ($\mu$) is the ratio of magnetic flux density (B) to the magnetising field strength (H).
$\mu =\frac{B}{H}$
In CGS ( centimeter-gram-second) dimension of B and H is same. Hence, magnetic permeability is dimensionless. But in SI unit, dimension of ​B and H is not same. Thus, permeability is not dimensionless.
Relative ​permeability is defined as the ratio of magnetic permeabity of any medium to the permeability of the vaccum. Hence, it is dimensionless. Thus, permeability and relative permeability have the same dimensions in CGS system.