If V is a vector space over the field F, the general linear group of V, written GL ( V ) or Aut ( V ), is the group of all automorphisms of V, i. e. the set of all bijective linear transformations V → V, together with functional composition as group operation.

If E is such a subfield, write Gal ( L / E ) for the group of field automorphisms of L that hold E fixed.

If E is a finite Galois extension of Q < sub > p </ sub > and the building is constructed from SL < sub > n </ sub >( E ) instead of SL < sub > n </ sub >( Q < sub > p </ sub >), the Galois group Gal ( E / Q < sub > p </ sub >) will also act by automorphisms on the building.

If X in addition belongs to some category, then the elements of G are assumed to act as automorphisms in the same category.

A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E, then E / F is a Galois extension, where F is the fixed field of G.

Most often it is used in the context of a manifold M. The mapping class group of M is interpreted as the group of isotopy-classes of automorphisms of M. So if M is a topological manifold, the mapping class group is the group of isotopy-classes of homeomorphisms of M. If M is a smooth manifold, the mapping class group is the group of isotopy-classes of diffeomorphisms of M. Whenever the group of automorphisms of an object X has a natural topology, the mapping class group of X is defined as Aut ( X )/ Aut < sub > 0 </ sub >( X ) where Aut < sub > 0 </ sub >( X ) is the path-component of the identity in Aut ( X ).

If M is an oriented manifold, Aut ( M ) would be the orientation-preserving automorphisms of M and so the mapping class group of M ( as an oriented manifold ) would be index two in the mapping class group of M ( as an unoriented manifold ) provided M admits an orientation-reversing automorphism.

If the field is algebraically closed of characteristic 0 and the algebra is finite dimensional then all Cartan subalgebras are conjugate under automorphisms of the Lie algebra, and in particular are all isomorphic.

If X is a Dynkin diagram, Chevalley constructed split algebraic groups corresponding to X, in particular giving groups X ( F ) with values in a field F. These groups have the following automorphisms:

If a litigant chooses to enforce a Federal right in a State court, he cannot be heard to object if he is treated exactly as are plaintiffs who press like claims arising under State law with regard to the form in which the claim must be stated -- the particularity, for instance, with which a cause of action must be described.

If, at any time during the assignment pass, the compiler finds that there are no more index words available for assignment, the warning message `` No More Index Words Available '' will be placed in the object program listing, the table will be altered to show that index words 1 through 96 are available, and the assignment will continue as before.

If the compiler finds that there are no more electronic switches available for assignment, the warning message `` No More Electronic Switches Available '' will be placed in the object program listing, the table will be altered to show that electronic switches 1 through 30 are available, and assignment will continue as before.

If this article / noun pair is used as the object of a verb, it ( usually ) changes to the accusative case, which entails an article shift in German – Ich sehe den Wagen.

If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their perpendicular height of incidence, i. e. their distance from the axis.

If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.

If, in an unsharp image, a patch of light corresponds to an object point, the center of gravity of the patch may be regarded as the image point, this being the point where the plane receiving the image, e. g., a focusing screen, intersects the ray passing through the middle of the stop.

If we look at this situation from far above, so that we cannot see the supporters, we see the large balloon as a small object animated by erratic movement.

If there be any party which is more pledged than another to resist a policy of restrictive legislation, having for its object social coercion, that party is the Liberal party.

If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

If a pixel-by-pixel ( image order ) approach to rendering is impractical or too slow for some task, then a primitive-by-primitive ( object order ) approach to rendering may prove useful.

If a rotating frame is chosen so that just the angular position of an object is held fixed, more complicated motion, such as elliptical and open orbits, appears because the centripetal and centrifugal forces will not balance.

If there is a direct object, the indirect object can be expressed by an object pronoun placed between the verb and the direct object.

If the average density ( including any air below the waterline ) of an object is less than water it will float in water and if it is more than water's it will sink in water.

If the obstructing object provides multiple, closely spaced openings, a complex pattern of varying intensity can result.

If a particular object did not support a, it could be easily added in the module.

Frege, however, did not conceive of objects as forming parts of senses: If a proper name denotes a non-existent object, it does not have a reference, hence concepts with no objects have no truth value in arguments.

If the occulting object has an atmosphere, however, some of the luminosity of the star can be refracted into the volume of the umbra.

If a physical object statement is to be translatable into a sense-data statement, the former must be at least deducible from the latter.

Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.

If a laser is swept across a distant object, the spot of laser light can easily be made to move across the object at a speed greater than c. Similarly, a shadow projected onto a distant object can be made to move across the object faster than c. In neither case does the light travel from the source to the object faster than c, nor does any information travel faster than light.

* If numbers have mean X, then.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If a detector was placed at a distance of 1 m, the ion flight times would be X and Y ns.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

* Theorem If X is a normed space, then X ′ is a Banach space.

If X ′ is separable, then X is separable.

If F is also surjective, then the Banach space X is called reflexive.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

If there is a bounded linear operator from X onto Y, then Y is reflexive.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.