Isomorphism under differentiation

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The discussion centers on whether the differentiation of the function sin(x) can be considered a valid cyclic group. A group requires a binary operation involving its elements, which leads to questions about the elements and operations in this context. Differentiation is identified as a function that operates on functions rather than a binary operation between two real-valued functions. While differentiation can be viewed cyclically when considering sin(x) and its derivatives, it lacks a direct connection to group theory. However, a cyclic group could potentially be defined using a set of functions like (sin(x), cos(x), -sin(x), -cos(x)) with consecutive differentiation as the operation.
tomgotthefunk
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Is 《sinx》under differentiation a valid cyclic group.
 
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A group must have a binary operation involving it's elements. What are the elements of the group you are asking about? What is the binary operation?

Perhaps you are thinking that differentiation operates on a set of real valued functions. That is true, but differentiation itself is not a real valued function. (Differentiation is a function from functions to other functions.) So differentiation of a function is not a binary operation involving two real valued functions.
 
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Consider sin as a row, then differentiation is cyclic, but there is not any connection with groups.
 
You could consider differentiation operations on the set (sinx, cosx, -sinx, -cosx) where the + operation is consecutive differentiation. With a little work, I am sure you could define a cyclic group.
 

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