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) |
Transformation group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transformation_group&oldid=32833