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 $m_g:A\to A$ 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 $g,h\in G$ $g\ne h$ we have $m_g\ne m_h$, in other words, the homomorphism $g\to m_g$ being injective, we say that the action is faithful.

Title	faithful group action
Canonical name	FaithfulGroupAction
Date of creation	2013-03-22 14:02:23
Last modified on	2013-03-22 14:02:23
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	8
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 16W22
Classification	msc 20M30