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November 24th, 2010, 05:21 AM  #1 
Member Joined: Nov 2010 Posts: 53 Thanks: 0  Question about def. of additive groups
Hi! Just a little question about the def. of additive groups. It just mean a group whose binary operation we do not know, but it is closed under addition righ? It does not mean that the groups binary operation necessarily is addition... 
November 24th, 2010, 05:56 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Question about def. of additive groups
It means a *group* whose operation we call "addition". So not only is it closed under the operation, but it is associative under the operation and there is an inverse for that operation.

November 24th, 2010, 06:49 AM  #3 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,767 Thanks: 5  Re: Question about def. of additive groups
Also, the notation is a bit different. Rather than writing a ? (a^1) = e (where e is the identity element), you can write a + (a) = 0 since the additive inverse of an element is its opposite/negative. 
November 24th, 2010, 06:55 AM  #4 
Senior Member Joined: Aug 2010 Posts: 195 Thanks: 5  Re: Question about def. of additive groups
It tends to be contextual as to whether a group is written additively or multiplicatively. By convention, groups are almost never written additively if they are not Abelian (commutative with respect to the operation). Strictly speaking however, the only difference is one of notation, as noted above.

November 24th, 2010, 07:57 AM  #5 
Member Joined: Nov 2010 Posts: 53 Thanks: 0  Re: Question about def. of additive groups
mhm, ok I see. Thanks for all the feedback.


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