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**dori1123** Let $\displaystyle p$ be a prime and let $\displaystyle G$ be a group of order $\displaystyle p^\alpha$. Prove that $\displaystyle G$ has a subgroup of order $\displaystyle p^\beta$, for every $\displaystyle \beta $ with $\displaystyle 0<=\beta<=\alpha$.

Hint: induction on $\displaystyle \alpha$ and use the theorem: If $\displaystyle p$ is a prime and $\displaystyle G$ is a group of prime power order $\displaystyle p^\alpha$ for some $\displaystyle \alpha>=1$, then $\displaystyle G$ has a nontrivial center.