History of Frobenius normal $p$-complement theorem

Question

It is well-known that Burnside’s and Frobenius’ normal $p$-complement theorems are the very foundations of modern theories relating to groups.

I wanted to sort out the history of theses theorems. In Wikipedia, we can find that the one due to Burnside can be located at section 243 of his book (1911, 2nd edition). However, Wikipedia does not give a clear reference addressing the resource of Frobenius’ theorem.

I looked up the theorem in several group theory books, but they always state it without citing any book or article of Frobenius. Any comment or answer that can help one obtain more clues is appreciated.

Answer 1

Frobenius' theorem appeared in his paper ["Über auflösbare Gruppen V", Sitzungsberichte der Könglich Preußischen Akademie der Wissenschaften 1901, 1324-1329]. Let me translate word-by-word the relevant statement:

If $p^\lambda$ is the highest power of a prime $p$, which divides the order of a group $G$, and if every element, which commutes with any subgroup of order $p^\mu$ and whose order is not divisible by $p$, commutes with every element of the subgroup, then G has a characteristic subgroup of index $p^\lambda$, which is formed by all elements of $G$, whose order is not divisible by $p$.

It is miracle how these guys got things down with so many words :-)