An isolic generalization of Cauchy's theorem for finite groups
In his note ``Applications of a simple counting technique'', \textit{M. Hausner} [Am. Math. Mon. 90, 127-129 (1983; Zbl 0509.10003)] stated the following principle: (H) ``Let X be a finite set, p a prime and f a mapping from X into X such that \(f^ p\) is the identity; then \(| X| \equiv | X_ 0| (mod p)\), where \(X_ 0\) is the set of fixed points of f.'' He then used (H) to prove Fermat's little theorem \(x^ p\equiv x(mod p)\) and Cauchy's theorem: ``If the prime p divides the order of a finite group G, then G has at least one element of order p.'' In the present paper (H) is generalized to: \((H^*)\) ``Let \(\alpha\) be an isolated set, p a prime and f a mapping with a partial recursive extension from \(\alpha\) into \(\alpha\) such that \(f^ p\) is the identity; then Req \(\alpha\) \(\equiv Req \alpha_ 0(mod p)\), where \(\alpha_ 0\) is the set of fixed points of f and Req \(\sigma\) the recursive equivalence type of \(\sigma\).'' The author then uses \((H^*)\) to generalize Fermat's little theorem (in its homogeneous form, i.e., as stated above) and Cauchy's theorem to isols.
scientific article; zbMATH DE number 4125389