Groups with $\pi$-minimal and $\pi$-layer minimal conditions. I

For an arbitrary set $\pi$ of prime numbers we study the properties and structure of groups satisfying the $\pi$-minimal and $\pi$-layer minimal conditions. In particular, we reveal the structure of almost $RN$-groups (and therewith locally solvable groups) and, assuming $2\in\pi$, also the structure of locally solvable (and therewith locally finite) groups with these conditions.