Let H be a normal subgroup of a finite group G and let a be an elementof G. Complete the following statement: The order of the elementaH in the factor group G/H is the smallest positive integer nsuch that an is .

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Q: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

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Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

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Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

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