An isolic generalization of Cauchy's theorem for finite groups

Summary

In his note [5] Hausner states a simple combinatorial principle, namely:

$$(H)\left\{ {\begin{array}{*{20}c} {if f is a function a non - empty finite set \sigma into itself, p a} \\ {prime, f^p = i_\sigma and \sigma _0 the set of fixed points of f, then } \\ {\left| \sigma \right| \equiv \left| {\sigma _0 } \right|(mod p).} \\\end{array}} \right.$$

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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.