Still another proof of the existence of Sylow-\(p\)-subgroups (Q1339980)

The classical inductive proof of the existence of Sylow \(p\)-subgroups became obsolete when \textit{H. Wielandt} published his ingenious non- inductive proof [Arch. Math. 10, 401-402 (1959; Zbl 0092.024)]. Since then there have been quite a few proofs given. The author gives another very interesting, simple yet inductive proof on the existence of Sylow \(p\)-subgroups using the fact that for every nontrivial subgroup \(U\) of a finite group \(G\) the set \(X(U) := \{f : G/U \to G\mid f(gU) \in gU\) for all \(g\in G\}\) of all systems of coset representatives of \(U\) in \(G\) is a fixed point free \(G\)-set of cardinality \(| X(U)| = | U|^{(G : U)}\) and hence, by orbit decomposition, there exists proper subgroups \(V_ i < G\) of \(G\) \((i = 1,\dots, n_ U)\) with \(| U|^{(G : U)} = \sum_ i (G : V_ i)\).