Isomorphism under differentiation

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Discussion Overview

The discussion revolves around the concept of isomorphism under differentiation, specifically whether the differentiation of the function sin(x) can be considered a valid cyclic group. The scope includes theoretical exploration of group properties in the context of differentiation and functions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether sin(x) under differentiation constitutes a valid cyclic group, prompting a discussion on the nature of groups and binary operations.
  • Another participant clarifies that a group requires a binary operation involving its elements and challenges the initial premise by noting that differentiation is a function from functions to functions, not a binary operation on real-valued functions.
  • A different viewpoint suggests that if sin(x) is considered as a row, differentiation could be cyclic, but this participant sees no connection to group theory.
  • Another participant proposes that differentiation operations on the set (sin(x), cos(x), -sin(x), -cos(x)) could be examined, suggesting that with further work, a cyclic group might be defined using consecutive differentiation as the operation.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between differentiation and group theory, with no consensus reached on whether sin(x) under differentiation can be classified as a cyclic group.

Contextual Notes

The discussion highlights the need for clarity on definitions and the nature of operations involved in group theory, particularly concerning the role of differentiation.

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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