  We prove that a square-integrable set-indexed stochastic process is a
set-indexed Brownian motion if and only if its projection on all the strictly
increasing continuous sequences are one-parameter $G$-time-changed Brownian
motions. In addition, we study the "sequence-independent variation" property
for group stationary-increment stochastic processes in general and for a
set-indexed Brownian motion in particular. We present some applications.
