Number of Subgroups of Prime Power Order is Congruent to 1 modulo Prime
Jump to navigation
Jump to search
This theorem requires a proof.
Similar to the First Sylow Theorem at first glance. The book suggests this proof is due to Frobenius, and is long.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
Theorem
Let $G$ be a finite group whose order is $n$.
Let $p$ be a prime number such that $p^k$ is a divisor of $n$.
Then the number of subgroups of order $p^k$ is congruent to $1$ modulo $p$.
Proof
Similar to the First Sylow Theorem at first glance. The book suggests this proof is due to Frobenius, and is long.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 59 \iota$