An isolic generalization of Cauchy's theorem for finite groups
type: Item
instance of: scholarly article
title: An isolic generalization of Cauchy's theorem for finite groups
zbMATH Open document ID: 0687.03025
DOI: 10.1007/BF01651326
author: J. C. E. Dekker
published in: Archive for Mathematical Logic
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03D50
zbMATH DE Number: 4125389
zbMATH Keywords: Fermat's little theorem
zbMATH Keywords: Cauchy's theorem
zbMATH Keywords: isols
MaRDI profile type: Publication
cites work: Congruences in isols with a finite modulus
cites work: Isols and Burnside's lemma
cites work: Q5661491
cites work: Applications of a Simple Counting Technique
cites work: Q3942950
cites work: Extensions to isols
description: scientific article; zbMATH DE number 4125389