In Lang's "Algebra", p.30-31, I'm confused about the order of reading the composition of two permutations. In p.30, it seems that we read it from left to right (see the bottom equations), but for p.31, we get the result [rki] by computing the product of four cycles from right to left, instead of from left to right. So, which kind of notation does Lang want to use?
2026-03-29 21:54:01.1774821241
About the notation of composition of permutations in Lang's book
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It seems you're misunderstanding the bottom of p. $30$. The operators are being applied right to left as usual:
$$\pi(\sigma)\pi(\tau)f=\pi(\sigma)\left(\pi(\tau)f\right)\;.$$
Perhaps what's confusing you is that $\pi(\sigma)$ is being applied to $\pi(\tau)f$ without first spelling out $\pi(\tau)f$? This is a choice in the order of spelling out, not in the order of application.