Is 《sinx》under differentiation a valid cyclic group.
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.
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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