A question regarding the conditions imposed on an index in the course of the proof of Sylow's theorem I.

abstract-algebra group-theory sylow-theory

You take the product over $k=0$ to $k=p^{\alpha-1}$. Therefore, if $k=p^{a}K$ for some $a\ge \alpha$ (and some integer $K$ with $p\nmid K$), then $p^a\mid k$, which means $k>p^{\alpha-1}$, a contradiction

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