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 March 31st, 2013, 11:19 AM #1 Newbie   Joined: Mar 2013 Posts: 2 Thanks: 0 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.
March 31st, 2013, 11:25 AM   #2
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Re: How to disprove these statements?

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 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.

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.

 April 1st, 2013, 02:18 AM #4 Senior Member   Joined: Sep 2008 Posts: 150 Thanks: 5 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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