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March 31st, 2013, 11:19 AM   #1
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How to disprove these statements?

*Every endomorphism is an epimorphism.

*Every two abelian groups of the same order are isomorphic.

*Every monomorphism is an isomorphism.

I know these statements are false, but I cannot find any example to disprove them. If you know, could you give me some example for these statements? Thanks.
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March 31st, 2013, 11:25 AM   #2
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Re: How to disprove these statements?

Quote:
Originally Posted by acelya
*Every endomorphism is an epimorphism.

*Every two abelian groups of the same order are isomorphic.

*Every monomorphism is an isomorphism.

I know these statements are false, but I cannot find any example to disprove them. If you know, could you give me some examples for these statements? Thanks.
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March 31st, 2013, 06:43 PM   #3
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Re: How to disprove these statements?

Quote:
Originally Posted by acelya
*Every endomorphism is an epimorphism.

*Every two abelian groups of the same order are isomorphic.

*Every monomorphism is an isomorphism.

I know these statements are false, but I cannot find any example to disprove them. If you know, could you give me some example for these statements? Thanks.
For 1 and 3, carefully write down the exact definitions and think about what they mean. Post them here so we can see what definitions you're working from. The category-theoretic definition of an epi, for example, is subtly different from the group-theoretical definition.

For #2, think six.
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April 1st, 2013, 02:18 AM   #4
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Re: How to disprove these statements?

Maybe one can give a few more hints:

1) Between any two abelian groups there is always a Homomorphism called the zero Homomorphism. Can you guess what is does?

2) There is only one abelian group of order 6 or of any order that is a square free number (i.e.: is not divisible by the square of a prime). The smallest number n, with more that one abelian group of order n is 4. ( To prove it: guess what the groups are and look at the order of each element in both groups.)

3) A monomorphism does not need to be an endomorphism. So you might want to look a a morphism from a 'small' group into a 'big' group. (small and big are not supposed to refer to any group theoretic terms, they are just to imagen, how an example should look like.)

I hope that helps. Tell me if it worked for you.
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