  In this paper, we investigate and discuss in detail the structures of
quaternion tensor SVD, quaternion tensor rank decomposition, and
$\eta$-Hermitian quaternion tensor decomposition with the isomorphic group
structures and Einstein product. Then we give the expression of the
Moore-Penrose inverse of a quaternion tensor by using the quaternion tensor
SVD. Moreover, we consider a generalized Sylvester quaternion tensor equation.
We give some necessary and sufficient conditions for the existence of a
solution to the generalized Sylvester quaternion tensor equation in terms of
the Moore-Penrose inverses of the quaternion tensors. We also present the
expression of the general solution to this tensor equation when it is solvable.
As applications of this generalized Sylvester quaternion tensor equation, we
derive some necessary and sufficient conditions for the existences of
$\eta$-Hermitian solutions to some quaternion tensor equations. We also provide
some numerical examples to illustrate our results.
