caa a22 4500 465743560 CHVBK 20180323111814.0 cr unu---uuuuu 170327e19901201xx s 000 0 eng 10.1007/BF01651326 doi (NATIONALLICENCE)springer-10.1007/BF01651326 Dekker J. Rutgers University, 08903, New Brunswick, NJ, USA aut An isolic generalization of Cauchy's theorem for finite groups [Elektronische Daten] [J. Dekker] Summary: In his note [5] Hausner states a simple combinatorial principle, namely: $$(H)\left\{ {\begin{array}{*{20}c} {if f is a function a non - empty finite set \sigma into itself, p a} \\ {prime, f^p = i_\sigma and \sigma _0 the set of fixed points of f, then } \\ {\left| \sigma \right| \equiv \left| {\sigma _0 } \right|(mod p).} \\\end{array}} \right.$$ . He then shows how this principle can be used to prove: (A) Fermat's little theorem, (B) Cauchy's theorem for finite groups, (C) Lucas' theorem for binomial numbers. Letε=(0,1, ...),ℱ 1−1 the family of all one-to-one functions from a subset ofε intoε andℳ 1−1 the family of all p.r. (i.e., partial recursive) one-to-one functions from a subset ofε intoε. A subsetα of ε is finite, ifα is not equivalent to a proper subset under a function inℱ 1−1. The setα is calledisolated, if it is not equivalent to a proper subset under a function inℳ 1−1. An isolated set can also be defined as a subset ofε which has no infinite r.e. (i.e., recursively enumerable) subset. While every finite set is isolated, there are $$c = 2^{\aleph _0 }$$ infinite sets which are isolated; these sets are calledimmune. It is the purpose of this paper to generalize (H) to a principle (H*) for isolated sets and to show how (H*) can be used to prove generalizations of Fermat's little theorem and Cauchy's theorem for finite groups. We have been unable to generalize Lucas' theorem in a similar manner. Springer-Verlag, 1990 Archive for Mathematical Logic Springer-Verlag 29/4(1990-12-01), 231-236 0933-5846 29:4<231 1990 29 153 https://doi.org/10.1007/BF01651326 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01651326 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Dekker J. Rutgers University, 08903, New Brunswick, NJ, USA aut NATIONALLICENCE 773 0- Archive for Mathematical Logic Springer-Verlag 29/4(1990-12-01), 231-236 0933-5846 29:4<231 1990 29 153 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer