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This was a question posed at the end of a problem sheet in a group theory class I am taking:

Problem

Let $G$ be any group, let $g$ be any element of $G$ of finite order, and let $p$ be any prime. Show that there exist $s, u \in G$ such that $g = su$ and each of the following holds:

  1. $|u|$ is a power of $p$;
  2. $p$ does not divide $|s|$; and
  3. $us = su$.

I am aware I have to provide some kind of constructive proof, however I am really not sure how to get a foot into the proof itself. It is not entirely intuitive which results I can leverage that would lead me to suitable $s,u$ that satisfy those three properties. I would very much appreciate some direction into this problem. Thank you.

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  • $\begingroup$ You could start by looking for examples in the symmetric group $S_4$ for various permutations and various primes. $\endgroup$ Commented Oct 31 at 21:18
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    $\begingroup$ Hint. Let $s$ and $u$ be powers of $g$. What is the order of $g^k$ in terms of $k$ and the order of $g$? $\endgroup$ Commented Oct 31 at 21:33

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