I've some questions regarding the following problem from Herstein. BTW I'm not looking for its solution:
- Do $\lambda_g$ is actually $\lambda_g(x)=xg$ when I write $x\lambda_g$ as $\lambda_g(x)?$
- For the convention I used do I need to show that $\lambda_{gh}=\lambda_g\lambda_h$ instead of $\lambda_{gh}=\lambda_h\lambda_g?$
By writing functions on the right of arguments, Herstein is encouraging you to compose them from left to right, so $\lambda_h\lambda_g$ means "do $\lambda_h$ first and $\lambda_g$ second".
You can instead write functions on the left of arguments (as is more common). If you still compose them from left to right, then you still have $\lambda_{gh}=\lambda_h\lambda_g$. However, when you write functions on the left, it is more natural to compose them from right to left. So then $\lambda_h\lambda_g$ ends up meaning "do $\lambda_g$ first and $\lambda_h$ second", so the true statement becomes $\lambda_{gh}=\lambda_g\lambda_h$.
Whatever you do, you shouldn't change the definition, so even writing functions on the left, you have $\lambda_g(x)=gx$. Then the answer to the second question depends on whether you compose functions from left to right or from right to left. If it's from left to right, like Herstein, you should prove $\lambda_{gh}=\lambda_h\lambda_g$, but if it's from right to left, as is more natural in your notation, you should prove $\lambda_{gh}=\lambda_g\lambda_h$. Both expressions are "the same", but they use different notation for the composition of $\lambda_g$ and $\lambda_h$.