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I would like to investigate a function that sends ##f(x)## to ##f(x) - \frac{1}{c}f(x^{c})##, or a function ##g## such that ##g(f(x)) = f(x) - \frac{1}{c}f(x^{c}).## Since symmetries produced by groups are used in physics, I thought there might be someone here who could help explain what the process is.

What I would like to know is that for at least some analytic function ##f(x)## that this transformation can form a Lie group in the constant ##c##, or that translating ##c## to ##c + t## keeps the solution manifold invariant. I would imagine I need to show continuity in some way. Then, how would I show such a manifold is smooth? How would I take a Lie derivative? Is there a convenient way I can satisfy the group axioms? A lot of online text is too scattered and hastily explained to give me concise insight into how to proceed.

It might be that instead of looking at a function ##g##, I should just start with ##f(x) - \frac{1}{c}f(x^{c})## and suppose there exists an operator ##L## (not necessarily linear) which sends ##f(x) - \frac{1}{c}f(x^{c})## to ##f(x) - \frac{1}{c+t}f(x^{c+t})##. I'm not really sure, I don't have formal training in Lie groups so I don't know what the setup should be.

What I would like to know is that for at least some analytic function ##f(x)## that this transformation can form a Lie group in the constant ##c##, or that translating ##c## to ##c + t## keeps the solution manifold invariant. I would imagine I need to show continuity in some way. Then, how would I show such a manifold is smooth? How would I take a Lie derivative? Is there a convenient way I can satisfy the group axioms? A lot of online text is too scattered and hastily explained to give me concise insight into how to proceed.

It might be that instead of looking at a function ##g##, I should just start with ##f(x) - \frac{1}{c}f(x^{c})## and suppose there exists an operator ##L## (not necessarily linear) which sends ##f(x) - \frac{1}{c}f(x^{c})## to ##f(x) - \frac{1}{c+t}f(x^{c+t})##. I'm not really sure, I don't have formal training in Lie groups so I don't know what the setup should be.

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