Let G be a permutation group acting on [n]={1,...,n} and
V={Viβ:i=1,...,n} be a system of n subsets of [n]. When
is there an element gβG so that g(i)βViβ for each iβ[n]? If
such g exists, we say that G has a G-marriage subject to V.
An obvious necessary condition is the {\it orbit condition}: for any β ξ =Yβ[n], βyβYβVyββYg={g(y):yβY} for some gβG. Keevash (J. Combin. Theory Ser. A 111(2005),
289--309) observed that the orbit condition is sufficient when G is the
symmetric group \Sym([n]); this is in fact equivalent to the celebrated
Hall's Marriage Theorem. We prove that the orbit condition is sufficient if and
only if G is a direct product of symmetric groups. We extend the notion of
orbit condition to that of k-orbit condition and prove that if G is the
alternating group \Alt([n]) or the cyclic group Cnβ where nβ₯4, then
G satisfies the (nβ1)-orbit condition subject to \V if and only if G
has a G-marriage subject to V