Number of Order p^(n-1) Subgroups in Order p^n Group

Theorem

Let $G$ be a group of order $p^n$.

Then the number of subgroups of order $p^{n-1}$ is congruent to $1$ modulo $p$.

Proof

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Sources

• 1895: Ferdinand Georg Frobenius: Verallgemeinerung des Sylow'schen Satzes ("A generalization of Sylow's theorem") (Sitzungsberichte K. Preuss. Akad. Wiss. Berlin pp. 981 – 993)