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August 2nd, 2010, 08:41 AM  #1 
Joined: Aug 2010 Posts: 2 Thanks: 0  groups...cancellation laws..
sup[pose a finite set G is closed under ans associative product and that both cancellation laws hold in G...prove that G must be a group...thnx in advance

August 2nd, 2010, 10:51 AM  #2 
Senior Member Joined: Aug 2010 Posts: 195 Thanks: 4  Re: groups...cancellation laws..
What have you tried? Where are you stuck? This is a very good question, it would be a shame to have the entire proof given away. As a starting point, what rules of a group still need to proven? 
August 4th, 2010, 01:44 PM  #3 
Joined: Aug 2010 Posts: 2 Thanks: 0  Re: groups...cancellation laws..
we need to look for the existence of identity element and then to proove that inverse exists...

August 4th, 2010, 05:10 PM  #4 
Senior Member Joined: Aug 2010 Posts: 195 Thanks: 4  Re: groups...cancellation laws..
Alright, so we need an identity and we need inverses. First thing is first, we need to know that there is an identity before we go looking for inverses. In the end, we are looking for an element which has no effect when I multiply it by any element, either on the left or on the right. But perhaps it is easier to take things step by step. For a given element , can we find a left identity? a right identity? If you would like, a hint: what can we say about the function defined by ? 

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Prove that a semigroup is a group in which both cancellation law holds,prove the left and right cancellation laws for a group g
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