  We derive a simple efficient algorithm for Abelian periods knowing all
Abelian squares in a string. An efficient algorithm for the latter problem was
given by Cummings and Smyth in 1997. By the way we show an alternative
algorithm for Abelian squares. We also obtain a linear time algorithm finding
all `long' Abelian periods. The aim of the paper is a (new) reduction of the
problem of all Abelian periods to that of (already solved) all Abelian squares
which provides new insight into both connected problems.
