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Suppose $G$ is a group with $|G|=30$. If $g \in G$ with $g \neq e$, $g^2=e$, then the permutation of $G$ defined by $x \mapsto gx$ apparently is a product of $15$ transpositions. Why would that be?

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  • $\begingroup$ What have you tried? $\endgroup$ – Dietrich Burde Jan 31 at 15:08
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For all $x\in G$, multiplication with $g$ swaps $x$ and $gx$. The $30$ elements come in $\frac{30}2$ pairs.

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HINT

Because $[G:\langle g\rangle]=\dfrac{|G|}{o(g)}=15$

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