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Sylow's Theorem Problems

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Verify that if H is a subgroup of G, and a ? G, then aHa^-1 is a subgroup of G.

Prove that if H is a finite subgroup of G, and a ? G, then |aHa^-1| = |H|. (Suggestion: The mapping h ? aha^-1 is one-to-one.)

Explain why if H is a Sylow p-subgroup of a finite group, then so is each conjugate of H.

Prove that if a finite group has only one Sylow p-subgroup for some prime p, then that subgroup must be normal.

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This solution explains how to solve problems related to Sylow's theorem.

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Clarification of Sylow's Theorem

A) Prove there is no simple group of order 200.
b) Assume that a group G has two Sylow p-subgroups K and H. Prove that K and H are isomorphic.
c) Show that a group G of order 2p^n has proper normal subgroup, where p is odd prime number and n > 0.

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