Group like operations that are not associative

In summary, a group can be defined by three properties: a set, an operation that is associative, an identity element exists, and every element has an inverse. The concept of an "almost-group" is introduced, where an operation fails the associativity test but meets the other two properties. The example of octonions is given as a possible almost-group. The discussion of coordinate transformations and their requirement to be associative is also mentioned. It is noted that while it may not be intuitively obvious, the associativity of coordinate transformations can be shown through function composition.
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pervect

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A group can be defined by the following three properties. (Source: wikipedia)

In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse.

Is there any example of an operation that fails the associativity test, but meets the other two tests? I'll refer to this hypothetical entity as an almost-group for the purposes of this post lacking any knowledge of a better name. While my specific motivation is in Lie groups, an example of such an operation that is a finite almost-group but not associative would be helpful.

The motivating question is just how intuitively obvious is it that coordinate transformations must form a Lie group. Intuitively, it's obvious that the identity transformation must exist, and the requirement that coordinate transformations be invertible also seems intuitive, but it's less intuitive why a coordinate transformation absolutely must be associative. That said, I can't think of any counterexamples that are not associative. I have some vague intuitive ideas why a coordinate transformation should be associative, but they're a bit fuzzy and I couldn't express them well.

While the motivational idea is about Lie "almost-groups", a discreete version would also be helpful to guide my intuition.

Note: an answer at the A level is acceptable, but I'd prefer that it be kept to the I level if at all possible. If there is an A-level answer readily available and it'd be too much work to dumb it down to the I-level, it'd be better to have the A-level answer than none at all.
 
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pervect said:
A group can be defined by the following three properties. (Source: wikipedia)
Is there any example of an operation that fails the associativity test, but meets the other two tests? I'll refer to this hypothetical entity as an almost-group for the purposes of this post lacking any knowledge of a better name. While my specific motivation is in Lie groups, an example of such an operation that is a finite almost-group but not associative would be helpful.

The motivating question is just how intuitively obvious is it that coordinate transformations must form a Lie group. Intuitively, it's obvious that the identity transformation must exist, and the requirement that coordinate transformations be invertible also seems intuitive, but it's less intuitive why a coordinate transformation absolutely must be associative. That said, I can't think of any counterexamples that are not associative. I have some vague intuitive ideas why a coordinate transformation should be associative, but they're a bit fuzzy and I couldn't express them well.

While the motivational idea is about Lie "almost-groups", a discreete version would also be helpful to guide my intuition.

Note: an answer at the A level is acceptable, but I'd prefer that it be kept to the I level if at all possible. If there is an A-level answer readily available and it'd be too much work to dumb it down to the I-level, it'd be better to have the A-level answer than none at all.
Say hello to the octonions:
https://en.wikipedia.org/wiki/Octonion
 
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pervect said:
... but it's less intuitive why a coordinate transformation absolutely must be associative. That said, I can't think of any counterexamples that are not associative. I have some vague intuitive ideas why a coordinate transformation should be associative, but they're a bit fuzzy and I couldn't express them well.
For any coordinate transformation, the "multiplication" is function composition ##(f \circ g)(\textbf{x}) = f(g(\textbf{x}))##. The associative law ##(f \circ g) \circ h = f \circ (g \circ h) ## follows by just plugging in the definition.
 
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