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Unformatted text preview: 5.4. GROUP ACTIONS AND GROUP STRUCTURE 253 Proof. We prove this statement by induction on n , the case n D 1 being Cauchys theorem. We assume inductively that G has a subgroup H of order p n 1 . Then OEG W H is divisible by p . Let H act on G=H by left multiplication. We know that OEG W H is equal to the number of fixed points plus the sum of the cardinalities of nonsingleton orbits. The size of every nonsingleton orbit divides the cardinality of H , so is a power of p . Since p divides OEG W H , and p divides the size of each nonsingleton orbit, it follows that p also divides the number of fixed points. The number of fixed points is nonzero, since H itself is fixed. Lets look at the condition for a coset xH to be fixed under left mul- tiplication by H . This is so if, and only if, for each h 2 H , hxH D xH . That is, for each h 2 H , x 1 hx 2 H . Thus x is in the normalizer of H in G (i.e., the set of g 2 G such that gHg 1 D H )....
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- Fall '08