Quick question about groups and their properties

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Discussion Overview

The discussion revolves around the properties of groups in abstract algebra, specifically addressing whether a set of elements that can yield multiple results from a binary operation can still be classified as a group. The conversation includes theoretical considerations and examples related to group operations and inverses.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions if a set can still be a group if two elements can produce more than one result, suggesting a need for clarification on the definition of group operations.
  • Another participant argues that if two elements produce different results (c ≠ d), then the operation does not satisfy the requirements of a group, as groups must yield a unique result for each pair of inputs.
  • A further explanation is provided regarding the necessity of unique inverses in groups, using the example of implicit functions to illustrate why multiple outputs would violate group properties.
  • Examples involving rotations of a cube are introduced to contextualize the discussion, noting that while rotations can yield different results, they still produce unique outcomes when applied in sequence.
  • It is mentioned that rotations are not commutative, yet they adhere to the group property of yielding unique results.

Areas of Agreement / Disagreement

Participants express differing views on the implications of multiple outputs in group operations. There is no consensus reached on the initial question regarding the classification of such sets as groups.

Contextual Notes

Participants reference specific mathematical properties and examples, but the discussion does not resolve the underlying assumptions about the definitions of operations and group properties.

jackscholar
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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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