Abstract Algebra I · Sylow theory and finite-group structure
The two Sylow counting constraints
Problem
Let $|G|=p^am$ with $p\nmid m$, and let $n_p$ be the number of Sylow $p$-subgroups. Assuming Sylow existence and conjugacy, prove $n_p\mid m$ and $n_p\equiv1\pmod p$.
Hint
Conjugation acts transitively on $\operatorname{Syl}_p(G)$, so $n_p=[G:N_G(P)]$.
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