  In this paper we study the fractional Dirichlet p-sub-Laplacian in a Haar
measurable set on homogeneous Lie groups. We prove fractional Sobolev and Hardy
inequalities and we also present a Lyapunov-type inequality for the fractional
p-sub-Laplacian. As a consequence of the Lyapunov-type inequality we show an
estimate of the first eigenvalue in a quasi-ball for the Dirichlet fractional
p-sub-Laplacian.
