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April 7th, 2012, 07:17 PM   #1
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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!
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April 7th, 2012, 08:10 PM   #2
uta
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Re: What does the / symbol mean?

Quote:
Originally Posted by IceVox121314
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?
The symbol "/" is used in group theory to denote the quotient group.

Quote:
Originally Posted by IceVox121314
what would the group Z^3/Z look like?
What is Z^3?
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April 9th, 2012, 04:00 AM   #3
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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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