I'm trying to read 'Uber die Charaktere der mehrfach transitiven Gruppen' written by Frobenius. There he mentioned some theorems of Netto.

I'm depending on the Google translator. and the translation reads

- If one multiplies the number of cycles of length $s$ that occur in all permutations of a group of order $h$ by the number $s$, one obtains a multiple of $h$, and if the group is $s$-fold transitive, the number $h$ itself.
- If you multiply the number of combinations of $x$ cycles of length 1, $y$ cycles of length 2, $z$ cycles of length 3, etc., which occur in all permutations of a group of order $h$, by the number $s=1^{x}x!2^{y}y!3^{z}z!\ldots$, you get a multiple of $h$, and if the group is $r=x+2y+3z+\cdots$-fold transitive, the number $h$ itself.

But the theorem does not make sense to me. Because for example if we take the cyclic subgroup of order 3 of $S_{3}$. Then $h=3$. If we take $s=2$, then I don't know how I should understand the theorem.

The paper of Frobenius is available in the website. https://www.e-rara.ch/doi/10.3931/e-rara-18850

Therefore, I want to ask

- whether the translation of the above theorem is correct.
- whether theorems similar to the one above are still significant in some branch of group theory. If so, I hope to know some recent paper or book.
- I especially want to understand the concept of $\textbf{dimension}$ of an irreducible representation $\chi^{\lambda}$ of the symmetric group $S_{n}$ associated to the partition $\lambda=(\lambda_{1}, \lambda_{2}, \ldots )$. The dimension is defined to be $n - \lambda_{1}$ and it is the smallest integer $r$ for which the character $\chi^{\lambda}$ appears as a constituent of the permutation character of $S_{n}$ on $[n]^{r}$, the $r$-fold cartesian product of the set of $n$-elements. Is there a reference for the proof of the equality of the two numbers?

I'm afraid that the question might be peripheral, but I hope to read the paper of Frobenius steadily even though it is old and in German.

Thank you very much !