  Let $G$ be a permutation group acting on $[n]=\{1, ..., n\}$ and
$\mathcal{V}=\{V_{i}: i=1, ..., n\}$ be a system of $n$ subsets of $[n]$. When
is there an element $g \in G$ so that $g(i) \in V_{i}$ for each $i \in [n]$? If
such $g$ exists, we say that $G$ has a $G$-marriage subject to $\mathcal{V}$.
An obvious necessary condition is the {\it orbit condition}: for any $\emptyset
\not = Y \subseteq [n]$, $\bigcup_{y \in Y} V_{y} \supseteq Y^{g}=\{g(y): y \in
Y \}$ for some $g \in G$. 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 $C_{n}$ where $n \ge 4$, then
$G$ satisfies the $(n-1)$-orbit condition subject to $\V$ if and only if $G$
has a $G$-marriage subject to $\mathcal{V}$.
