Proof of P-group Theorem (1)

proofp-groups

Theorem

A group $(G,\cdot )$ is a p-group if and only if $|G|=p^k$ for some $k$.

Proof

If $(G,\cdot )$ is a p-group, and assume $\exists g\ne p$ prime, where $g\mid n=|G|$.

Cauchy’s Theorem $⟹\exists g\in G,{\mathrm{ord}}_{}(g)=g$ not a power of $p$ (contradiction).

Conversely, if $|G|=p^k$, $g\in G⟹{\mathrm{ord}}_{}(g)\mid |G|=p^k⟹$ by Lagrange’s Theorem that ${\mathrm{ord}}_{}(g)$ is a power of $p$. Thus, $G$ is a p-group. $\square$

Remark

If $(G,\cdot )$ is commutative with $n=p_1^{{\alpha }_1}\cdot \dots \cdot p_r^{{\alpha }_r}$, $p_i$ distinct primes, $(G,\cdot )$ admits internal direct product decomposition as product of $p_i$-group $G[p_i]\le G$, also commutative.

See

• Proof of P-group Theorem (2)