A Generalization of a Result of Sinnott 2272

  • H . KISILEVSKYCorollary
  • Published 1997


To the memory of Olga Taussky-Todd, a friend, a collegue and an inspiration 1. Sinnott's Theorem. Let p be a prime number and suppose that ? is a prop group isomorphic to Z p ; the additive group of p-adic integers. For each integer n; let ? n = p n ? and G n = ?=? n : Let A be a discrete ?-module and deene A n = A ?n = fa 2 A j (a) = a for all 2 ? n g: Then A = A n : Proposition 1. If A n is nite for all n; then jA n+1 j jA n j (mod p n+1): Proof. A n+1 is a nite G n+1-module so that A n+1 = B C where B is the set of those elements in A n+1 not xed by any non-trivial element of G n+1 , and C = A n+1 n B: Since G n+1 is a cyclic group it follows that every element of C is xed by the the subgroup of order p in G n+1 ; and so C A ?n = A n : The opposite inclusion is clear so C = A n : Counting we have, jA n+1 j = jBj + jA n j: Since B is a union of orbits each of which contains p n+1 elements it follows that jA n+1 j jA n j (mod p n+1): 225

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@inproceedings{KISILEVSKYCorollary1997AGO, title={A Generalization of a Result of Sinnott 2272}, author={H . KISILEVSKYCorollary}, year={1997} }