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An algebraic group $$k$$ acting regularly on an algebraic variety $$V$$. More precisely, it is a triplet $( G , V , \tau )$ where $$\tau : G \times V \rightarrow V$$ ($\tau ( g , x ) = g x$) is a morphism of algebraic varieties satisfying the conditions: $e x = x$, $g ( h x ) = ( g h ) x$ for all $x \in V$ and $g , h \in G$ (where $E$ is the unit of $$k$$). If $G , V$ and $$\pi$$ are defined over a field $$k$$, then $( G , V , \tau )$ is called an algebraic group of $$k$$-transformations. For instance, $( G , G , \tau )$, where $$\pi$$ is the adjoint action or an action by shifts, is an algebraic group of transformations. If $$k$$ is an algebraic subgroup in $GL ( n )$ and $$\pi$$ is its natural action on the affine space $V = k ^ { n }$, then $( G , V , \tau )$ is an algebraic group of transformations. For each point $x \in V$ one denotes by $G ( x ) = \{ g x : g \in G \}$ the orbit of $$\pi$$, and by $$G _ { X } = \{ g \in G : g x = x \}$$ the stabilizer of $$\pi$$. The orbit $G ( x )$ need not necessarily be closed in $$V$$, but closed orbits exist always, e.g. orbits of minimal dimension are closed. An algebraic group of transformations is sometimes understood to mean a group $$k$$ which is acting rationally (but not necessarily regularly) on an algebraic variety $$V$$ (this means that $$\tau : G \times V \rightarrow V$$ is a rational mapping, and the above properties of $$\pi$$ are valid for ordinary points). It was shown by A. Weil [[#References|[3]]] that there always exists a variety $$V ^ { 1 }$$, birationally isomorphic to $$V$$, and such that the action of $$k$$ on $$V ^ { 1 }$$ induced by the rational action of $$k$$ on $$V$$ is regular. The problem of describing the orbits, stabilizers, fields of invariant rational functions (cf. [[Invariants, theory of|Invariants, theory of]]), and of constructing quotient varieties are fundamental in the theory of algebraic groups of transformations and have numerous applications.

====References====
<table><tr><td valign="top">[1]</td> <td valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </td></tr><tr><td valign="top">[2a]</td> <td valign="top">  J.A. Dieudonné,   "Invariant theory: old and new" , Acad. Press  (1971)  {{MR|0279102}} {{ZBL|0258.14011}} </td></tr><tr><td valign="top">[2b]</td> <td valign="top">  D. Mumford,   "Geometric invariant theory" , Springer  (1965)  {{MR|0214602}} {{ZBL|0147.39304}} </td></tr><tr><td valign="top">[2c]</td> <td valign="top">  D. Mumford,   "Projective invariants of projective structures and applications" , ''Proc. Internat. Congress mathematicians (Stockholm, 1962)'' , Inst. Mittag-Leffler  (1963)  pp. 526–530  {{MR|0175899}} {{ZBL|0154.20702}} </td></tr><tr><td valign="top">[2d]</td> <td valign="top">  C.S. Seshadri,   "Quotient spaces modulo reductive groupes and applications to moduli of vector bundles on algebraic curves" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 479–482      {{MR|}} {{ZBL|}} </td></tr><tr><td valign="top">[3]</td> <td valign="top">  A. Weil,   "On algebraic groups and homogeneous spaces"  ''Amer. J. Math.'' , '''77''' :  2  (1955)  pp. 355–391  {{MR|0074084}} {{ZBL|0065.14202}} </td></tr></table>



====Comments====
The notion in question is also called an algebraic transformation space.