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Sylow's Theorem

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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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Solution Summary

This solution provides a step-by-step explanation of how to solve the given problem involving Sylow's Theorem.

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a) Proof:
Since the group G has order 200 = 2^3 * 5^2, we consider its Sylow 5-subgroup. According to Sylow's
Theorem, the number of its Sylow 5-subgroup is 5k+1 | 2^3 = 8. Then the only possibility is k = 0.
So G has unique Sylow 5-subgroup and hence it must be ...

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