Let \(\mathcal O\) be a complete Noetherian local ring with residue field \(k\) of characteristic \(p>0\), and \(P\) a \(p\)-group. An \(\mathcal O\)-algebra of the form \(A=\text{End}_{\mathcal O}(M)\), where \(M\) is an endopermutation \({\mathcal O}P\)-lattice having an indecomposable direct summand with vertex \(P\), is called a Dade \(P\)-algebra. Two Dade \(P\)-algebras are called similar, if \(A\otimes B^{op}\simeq\text{End}_{\mathcal O}(M)\) for a permutation \({\mathcal O}P\)-lattice \(M\) having \(\mathcal O\) as a direct summand. The tensor product induces a group structure on the set \(D_{\mathcal O}(P)\) of similarity classes of Dade \(P\)-algebras; \(D_{\mathcal O}(P)\) is called the Dade group of \(P\). If \(Q\) is a subgroup of \(P\), \(R\) a quotient of \(P\) and \(P\to P'\) is a group isomorphism, there are functors inducing maps between Dade groups: restriction \(\text{Res}^P_Q\colon D_{\mathcal O}(P)\to D_{\mathcal O}(Q)\), inflation \(\text{Inf}^P_R\colon D_{\mathcal O}(R)\to D_{\mathcal O}(P)\), deflation \(\text{Def}^P_R\colon D_{\mathcal O}(P)\to D_{\mathcal O}(R)\), tensor induction \(\text{Ten}^P_Q\colon D_{\mathcal O}(Q)\to D_{\mathcal O}(P)\), and isomorphism \(\text{Iso}^{P'}_P\colon D_{\mathcal O}(P)\to D_{\mathcal O}(P')\). In this paper, the author investigates the effect of these operations on the relative syzygies \(\Omega_X(\mathcal O)\), which is defined as the kernel of the augmentation map \({\mathcal O}X\to\mathcal O\), where \(X\) is a \(P\)-set. The main result of the paper is a formula expressing \(\text{Ten}^P_Q(\Omega_X(\mathcal O))\) in terms of elements of the Dade group of \(P\). If \(\Omega_X(\mathcal O)\) is an endopermutation \({\mathcal O}P\)-lattice, define \(\Omega_X=\text{End}_{\mathcal O}\Omega_X(\mathcal O)\), otherwise let \(\Omega_X=0\). Then the above mentioned formula implies that the subgroup of \(D_{\mathcal O}(P)\) generated by the relative syzygies \(\Omega_X\) is invariant under the five operations. In the final part of the paper, the author uses these techniques to investigate the structure of \(D_{\mathcal O}(P)\). In particular, he obtains an alternative proof of the main result of \textit{S.~Bouc} and \textit{J.~Thévenaz} [Invent. Math. 139, No. 2, 275-349 (2000; see the preceding review Zbl 0954.20002)], and characterizations of the situation when \(\Omega_X\) is a torsion element of \(D_{\mathcal O}(P)\).