Detection of some elements in the stable homotopy groups of spheres

In this paper we constructs a new nontrivial family in the stable homotopy groups of spheres ${\pi }_{p^{n}q+2pq+q-3}S$ which is of order $p$ and is represented by $k_0{h}_n\in Ex{t}_A^{3,p^{n}q+2pq+q}({\mathbb{Z}}_p,{\mathbb{Z}}_p)$ in the Adams spectral sequence, where $p\ge 5$ is an odd prime, $n\ge 3$ and $q=2(p-1)$. In the course of the proof, a new family of homotopy elements in ${\pi }_{\ast }V(1)$ which is represented by $\beta_{\ast}{i^{\prime}}{\ast}i{\ast}({h}_n)\in Ext_A^{2,p^nq+(p+1)q+1}(H^{\ast}V(1),$\mathbb{Z}$_p)$ in the Adams sequence is detected.