# 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 $g\mapsto m_{g}$.

If for any pair $g,h\in G$ $g\neq h$ we have $m_{g}\neq m_{h}$, in other words, the homomorphism $g\to m_{g}$ being injective, we say that the action is faithful.

Title faithful group action FaithfulGroupAction 2013-03-22 14:02:23 2013-03-22 14:02:23 rspuzio (6075) rspuzio (6075) 8 rspuzio (6075) Definition msc 16W22 msc 20M30