There has been considerable interest in the methods of deriving congruences from group actions; see \textit{G.-C. Rota} and \textit{B. Sagan} [Eur. J. Comb. 1, 67-76 (1980; Zbl 0453.05008)], or \textit{B. Sagan} [J. Number Theory 20, 210-237 (1985; Zbl 0577.10003)]. The author introduces a different approach to this question. Let \(p\) be a prime and \(H\) a finite set. A wheel system on \(H\) is a family \(\mathcal W\) of subsets of \(H\) (called wheels) with the following three properties: (1) the order of any wheel is a power of \(p\) (a wheel of order \(p^ i\) is called a \(p^ i\)-wheel), (2) distinct wheels of the same order are disjoint, and (3) each \(p^{i+1}\)-wheel is the union of \(p^ i\)-wheels. Wheels of order \(>1\) are called proper. A wheel system \(\mathcal W\) is called complete if \(\mathcal W\) is maximal with respect to the conditions (1), (2) and (3). When one assigns to each proper \(p^ i\)-wheel of a given wheel system \(\mathcal W\) a cyclic ordering of the \(p^{i-1}\)-wheels it contains, \(\mathcal W\) gets an orientation. A spinning is such a permutation of the elements of a given proper wheel that takes its subwheels and preserves its orientation. The subgroup generated by all spinnings of a complete wheel system is the symmetric group \({\mathcal S}_ H\). After proving numerous interesting technical lemmas the author gives some number theoretic applications of his approach. So, he proves (Lemma 3.5): Let \(a= b_ 1+ \cdots+ b_ k\) and suppose \((a,p)= (a,b_ 1,\dots,b_ k)=1\). Then \[ \begin{pmatrix} ap^ n\\ b_ 1 p^ n,\ldots,b_ k p^ n\end{pmatrix}\equiv 0\pmod a. \] Also, he proves the following result (Theorem 5.1): Let \(I(m)\) denote the number of idempotent maps \({\mathbf m}\to {\mathbf m}\). If \(p>2\), then \[ I(p^ n)\equiv 1+ p^{n-1} \left[(\mu-n) p(p-1)+sp+\sum^{p-1}_{k=1} {p\choose k} \sum^ s_{j=0} k^{p+ j-k}{s\choose j}\right]\pmod{p^{n+1}}, \] where \(s=(p-1)(n-1)\) and \(\mu\) denotes the number of proper wheels of a wheel system on \(p^ n\) elements. Concerning some divisibility properties of the Stirling numbers of second kind he proves (Theorem 5.2): Let \(P(m,k)\) denote the number of surjections \({\mathbf m}\to {\mathbf k}\). Then \[ P(p^ n,k)\equiv P(p^{n-1},k)+ p^{n-1} \bigl[P(p+(n-1)(p-1),k)-P(1+ (n-1)(p-1),k)\bigr]\pmod{p^{n+1}}. \]