Transformation group

A permutation group $(G;M)$ acting on a set $M$. If, moreover, $M$ is endowed with a certain structure and if the elements of $G$ preserve this structure, then one appropriately says that $G$ is a transformation group of this structure. The name of the transformation group reflects to a certain extent the name of the structure endowed on $M$. E.g. if $M$ is a vector space over a skew-field, then groups preserving this structure are called linear groups (cf. Linear group). Moreover, automorphism groups of modules over various rings are often called linear groups. In particular, if $M$ is a free finite-dimensional module over the ring $\mathbf Z$ of integers, one speaks of crystallographic groups (cf. Crystallographic group). If $M$ is a topological space and $G$ consists of the automorphisms of $M$, one speaks of continuous transformation groups. If $M=K$ is a field and $G$ is a finite automorphism group of $K$, then $G$ is the Galois group of the extension $K/L$, where $L$ is the subfield of elements that are fixed under the action by elements of $G$. One can also consider situations in which $G$ and $M$ are endowed with structures of the same type and where the action of $G$ on $M$ is a morphism in the corresponding category. E.g. if $G$ is a topological group continuously acting on a topological space $M$, one speaks of a topological transformation group (cf. Topological group); Lie transformation groups and algebraic transformation groups are defined similarly (cf. Lie transformation group; Algebraic group of transformations).

References

[1]	A.I. [A.I. Mal'tsev] Mal'cev, "Groups and other algebraic systems" A.D. Aleksandrov (ed.) A.N. Kolmogorov (ed.) M.A. Lavrent'ev (ed.) , Mathematics, its content, methods and meaning , 3 , Amer. Math. Soc. (1962) pp. Chapt. 20 (Translated from Russian)