View source for Transformation group

A [[Permutation group|permutation group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t0937301.png" /> acting on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t0937302.png" />. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t0937303.png" /> is endowed with a certain structure and if the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t0937304.png" /> preserve this structure, then one appropriately says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t0937305.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t0937306.png" />. E.g. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t0937307.png" /> is a vector space over a skew-field, then groups preserving this structure are called linear groups (cf. [[Linear group|Linear group]]). Moreover, automorphism groups of modules over various rings are often called linear groups. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t0937308.png" /> is a free finite-dimensional module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t0937309.png" /> of integers, one speaks of crystallographic groups (cf. [[Crystallographic group|Crystallographic group]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373010.png" /> is a topological space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373011.png" /> consists of the automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373012.png" />, one speaks of continuous transformation groups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373013.png" /> is a field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373014.png" /> is a finite automorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373016.png" /> is the Galois group of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373018.png" /> is the subfield of elements that are fixed under the action by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373019.png" />. One can also consider situations in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373021.png" /> are endowed with structures of the same type and where the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373023.png" /> is a morphism in the corresponding category. E.g. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373024.png" /> is a topological group continuously acting on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093730/t09373025.png" />, one speaks of a topological transformation group (cf. [[Topological group|Topological group]]); Lie transformation groups and algebraic transformation groups are defined similarly (cf. [[Lie transformation group|Lie transformation group]]; [[Algebraic group of transformations|Algebraic group of transformations]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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) {{MR|}} {{ZBL|}} </TD></TR></table>