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April 7th, 2012, 07:17 PM  #1 
Newbie Joined: Apr 2012 Posts: 4 Thanks: 0  What does the / symbol mean?
Hey, I know this is a stupid question, but I've searched around and I don't know what the / symbol is used to denote in group theory. Is it similar to set minus? For example, what would the group Z^3/Z look like? thanks in advance! 
April 7th, 2012, 08:10 PM  #2  
Newbie Joined: Feb 2012 Posts: 18 Thanks: 0  Re: What does the / symbol mean? Quote:
Quote:
 
April 9th, 2012, 04:00 AM  #3 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88  Re: What does the / symbol mean?
normally, something like Z^3 is shorthand for Z x Z x Z, or triples of integers. we can make this a group by setting: (a,b,c) + (a',b',c') = (a+a',b+b',c+c'). literally, Z^3/Z makes no sense, as Z is not a subgroup of Z^3. however, what may be meant is the subgroup Z x {0} x {0}, which is isomorphic to it. this has the effect of essentially "setting one coordinate to 0" (in this case, the first coordinate). an element of Z^3/(Z x {0} x {0}) looks like this (0,b,c) + ((Z x {0} x {0}), but a common abuse of notation is to "drop the subgroup at the end", and write something like: Z^3/Z ? Z^2 sending (a,b,c)>(b,c). ********* i would check your text carefully, though, as a small typo in what you wrote might change the meaning entirely. 

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