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HallsofIvy

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Okay, what have you tried? This really only involves verifying the definitions of "homomorphism" and "subgroup".

The only condition on a "homomorphism" is that it maintain group operations. If f and g are two differentiable functions, is it true that theta(f+ g)= theta(f)+ theta(g)?

Since saying that C1 is a group means that things like associativity of the operation is true, we don't have to reprove those for operations in the subgroup. We only need to show

(1) the 0 element is in the subset

(2) the subgroup is "closed" under the group operation

The 0 element of this group is the 0 function, f(x)= 0 for all x. Is that in this subset?

Suppose f and g are in this subset. Is f+ g also in this subset? That is, is (f+ g)(0)= 0?

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