  We introduce the concept of group state transfer on graphs, summarize its
relationship to other concepts in the theory of quantum walks, set up a basic
theory, and discuss examples.
  Let $X$ be a graph with adjacency matrix $A$ and consider quantum walks on
the vertex set $V(X)$ governed by the continuous time-dependent unitary
transition operator $U(t)= \exp(itA)$. For $S,T\subseteq V(X)$, we says $X$
admits "group state transfer" from $S$ to $T$ at time $\tau$ if the submatrix
of $U(\tau)$ obtained by restricting to columns in $S$ and rows not in $T$ is
the all-zero matrix. As a generalization of perfect state transfer, fractional
revival and periodicity, group state transfer satisfies natural monotonicity
and transitivity properties. Yet non-trivial group state transfer is still
rare; using a compactness argument, we prove that bijective group state
transfer (the optimal case where $|S|=|T|$) is absent for almost all $t$.
Focusing on this bijective case, we obtain a structure theorem, prove that
bijective group state transfer is "monogamous", and study the relationship
between the projections of $S$ and $T$ into each eigenspace of the graph.
  Group state transfer is obviously preserved by graph automorphisms and this
gives us information about the relationship between the setwise stabilizer of
$S\subseteq V(X)$ and the stabilizers of naturally defined subsets obtained by
spreading $S$ out over time and crudely reversing this process. These
operations are sufficiently well-behaved to give us a topology on $V(X)$ which
is likely to be simply the topology of subsets for which bijective group state
transfer occurs at that time. We illustrate non-trivial group state transfer in
bipartite graphs with integer eigenvalues, in joins of graphs, and in symmetric
double stars. The Cartesian product allows us to build new examples from old
ones.
