Transformation group

A permutation group acting on a set . If, moreover, is endowed with a certain structure and if the elements of preserve this structure, then one appropriately says that 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 . E.g. if 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 is a free finite-dimensional module over the ring of integers, one speaks of crystallographic groups (cf. Crystallographic group). If is a topological space and consists of the automorphisms of , one speaks of continuous transformation groups. If is a field and is a finite automorphism group of , then is the Galois group of the extension , where is the subfield of elements that are fixed under the action by elements of . One can also consider situations in which and are endowed with structures of the same type and where the action of on is a morphism in the corresponding category. E.g. if is a topological group continuously acting on a topological space , 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)