In fact, assume that S is a set, G a group of transformations in S and ( G, s ) a fuzzy subgroup of G . Then, by setting
12.
The fundamental property that singles out all projective geometries is the " elliptic " group of transformations can move any line to the " line at infinity " ).
13.
The term " representation of a group " is also used in a more general sense to mean any " description " of a group as a group of transformations of some mathematical object.
14.
A group of transformations of X, to be denoted by G, is a set of ( measurable ) 1 : 1 and onto transformations of X into itself, which satisfies the following conditions:
15.
The word " symmetry " in the above statement refers more precisely to the covariance of the form that a physical law takes with respect to a one-dimensional Lie group of transformations satisfying certain technical criteria.
16.
Suppose we have a " flow ", i . e . the generator of a smooth one-dimensional group of transformations of the configuration space, which maps on-shell states to on-shell states while preserving the boundary conditions.
17.
Also, the above expression is a quadratic form of signature ( 3, 1 ) on spacetime, and the group of transformations which leaves this quadratic form invariant is the indefinite orthogonal group O ( 3, 1 ), a Lie group.
18.
Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group.
19.
In the seminal paper which introduced categories, Saunders Mac Lane and S . Eilenberg stated : " This may be regarded as a continuation of the Klein Erlanger Programm, in the sense that a geometrical space with its group of transformations is generalized to a category with its algebra of mappings"
20.
In mathematics, "'monodromy "'is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they'run round'a group of transformations acting on the data that encodes what does happen as we'run round '.
