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# faithful group action

Let $A$ be a $G$-set, that is, a set acted upon by a group $G$ with action $\psi:G\times A\to A$. Then for any $g\in G$, the map $m_{g}\colon A\to A$ defined by

$m_{g}(x)=\psi(g,x)$ |

is a permutation of $A$ (in other words, a bijective function from $A$ to itself) and so an element of $S_{A}$. We can even get an homomorphism from $G$ to $S_{A}$ by the rule $g\mapsto m_{g}$.

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## Mathematics Subject Classification

16W22*no label found*20M30

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