Quick question about groups and their properties

[jackscholar]

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?

 

[tiny-tim]

hi jackscholar!

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

 

[chiro]

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.

 

[jackscholar]

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.

 

[chiro]

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.

 

[jackscholar]

Thank you both for your help. I highly appreciate it.