Prove image of symmetric group into additive group of real numbers is zero Suppose $ f : S_{n} \rightarrow (\mathbb{R} , + , 0 , - )$ is a group homomorphism. Prove $ f(S_{n}) = {0} $, i.e., $f(\sigma) = 0$ for every $ \sigma \in S_{n}$with $n \geq 1 $
I cannot seem to find the reason why the transpositions ( $ S_{2} $ ) should have this property. This would of course immediately solve it. Help would be much appreciated!!

Every element of the additive group of real numbers has infinite order except the identity. Elements of finite order, such as all elements in a finite group, can only be mapped to elements of finite order. The result follows.
