Quick question about groups and their properties

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In a group, each pair of elements must produce a unique result under the group operation, meaning if two elements yield multiple outcomes, it does not satisfy the group properties. Groups require that every element has a unique inverse, which is compromised if multiple results arise from the same operation. The discussion highlights that while implicit functions can produce multiple values, groups cannot function that way. An example involving rotations of a cube illustrates that even non-commutative operations yield unique results, reinforcing the necessity for uniqueness in group operations. Therefore, if multiple answers arise from the operation of two group elements, it cannot be classified as a group.
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If two elements in a group operate together and can create more then 1 answer (this answer is still a part of set, not foreign) is it still a group, if so why?
 
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hi jackscholar! :smile:
jackscholar said:
If two elements in a group operate together and can create more then 1 answer (this answer is still a part of set, not foreign) is it still a group, if so why?

you mean, if ab = c and ab = d, and c ≠ d ?

then your "=" is not even an operation
 
jackscholar said:
If two elements in a group operate together and can create more then 1 answer (this answer is still a part of set, not foreign) is it still a group, if so why?

Groups need to create unique answers for each pair of inputs into the binary operation.

You can have these kinds of scenarios with implicit functions like when you have say x^2 + y^2 = 1, where you can multiple y values or x values for a given x or y but in groups you can't have this happen.

If you want another reason why, groups have to have inverse elements and if you had two possible cases, then you couldn't have a unique inverse.

Again if you are wondering about inverses consider the function y = x^2 against y = e^x and think about where finding the inverse fails for y = x^2 vs e^x and that will give you a visual reason you can't have your situation cause groups must have an inverse element for every group element in the group.
 
In this case it is rotations of a cube and there needed to be a rotation added. When a 90 degree rotation of the y-axis was added it created various different results when different verticies of said cube were put under say, a x-y plane reflection followed by a 90 degree rotation of the y axis. This created the identity, as opposed to a different point which didn't.
 
jackscholar said:
In this case it is rotations of a cube and there needed to be a rotation added. When a 90 degree rotation of the y-axis was added it created various different results when different verticies of said cube were put under say, a x-y plane reflection followed by a 90 degree rotation of the y axis. This created the identity, as opposed to a different point which didn't.

Rotations in general are not-commutative (i.e. AB <> BA in general), but applying rotations will always give a unique answer just so you can put this into context for your original question.
 
Thank you both for your help. I highly appreciate it.
 

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