Order of Finite p-Group is Power of p
Let $G$ be a finite group.
Let $p$ be a prime number.
Then $G$ is a $p$-group.
Aiming for a contradiction, suppose:
- $\order G = k p^n: p \nmid k$
where $\order G$ denotes the order of $G$.
- $k \nmid p^n$
From the First Sylow Theorem:
- $\exists H \le G: \order H = k$
where $H \le G$ denotes that $H$ is a subgroup of $G$.
- $\exists h \in H: \order h \divides k \implies \order h \nmid p$
where $\divides$ denotes divisibility.
- $\exists h \in G: \order h \ne p^n: n \in \Z$