The eigensolutions of the multigroup neutron transport equation for the isotropic scattering medium in plane geometry are found by developing Case's method. The solution of the adjoint equation is also given in an explicit form. The completeness of the eigensolutions is proved by using the orthogonality relation to the adjoint solution. The degeneracy for the continuous eigenvalue is expressed in two ways: the unit vector system and the eigenvector system. In the latter case the orthogonality is formed, and the eigenvector system is thought of as a proper normal mode. As an application of this method, the infinite medium Green's function is constructed.