March 30, 2017

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I have two questions... I would really appreciate some help...

1) Suppose phi is a homomorphism from G to G' and that G' has an element of order 8. Prove that G has an element of order 8.

(I can show that G has an element g^8 that maps to the identity in G', but I think this shows only that g^8 is in the kernal of G... I'm stuck on how to show that it has an element of order 8)

2) Suppose G is a cyclic group and that 6 divides the order of G. How many elements of order 6 does G have?

(I have no idea what to do with this... wouldn't it depend on how large G is?)

any help or hints are very appreciated! thanks!

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