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March 31st, 2013, 12:19 PM  #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, 12:25 PM  #2  
Newbie Joined: Mar 2013 Posts: 2 Thanks: 0  Re: How to disprove these statements? Quote:
 
March 31st, 2013, 07:43 PM  #3  
Senior Member Joined: Aug 2012 Posts: 2,137 Thanks: 623  Re: How to disprove these statements? Quote:
For #2, think six.  
April 1st, 2013, 03:18 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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