Summary
In his note [5] Hausner states a simple combinatorial principle, namely:
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He then shows how this principle can be used to prove:
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(A)
Fermat's little theorem,
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(B)
Cauchy's theorem for finite groups,
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(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.
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References
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Dekker, J.C.E. An isolic generalization of Cauchy's theorem for finite groups. Arch Math Logic 29, 231–236 (1990). https://doi.org/10.1007/BF01651326
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DOI: https://doi.org/10.1007/BF01651326