The set of all automorphisms of an object forms a group, called the automorphism group.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

The set of all automorphisms is a subset of End ( X ) with a group structure, called the automorphism group of X and denoted Aut ( X ).

The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group < var > G </ var >, denoted by Aut (< var > G </ var >), forms itself a group, the automorphism group of < var > G </ var >.

In other words, an automorphism of E / F is an isomorphism α from E to E such that α ( x ) = x for each x in F. The set of all automorphisms of E / F forms a group with the operation of function composition.

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.

Just as O ( n ) is the group of automorphisms which keep the quadratic polynomials x < sup > 2 </ sup > + y < sup > 2 </ sup > + ... invariant, F < sub > 4 </ sub > is the group of automorphisms of the following set of 3 polynomials in 27 variables.

F < sub > 4 </ sub > is the only exceptional lie group which gives the automorphisms of a set of real commutative polynomials.

In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary ( or antiunitary ) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.

An important example of this is when elements of G act by conjugation, so that the set of operators consists of the inner automorphisms.

Note that automatically G acts by automorphisms ( bijections ) on the set.

into the group of automorphisms of the object X in the category C. The pair ( X, ρ ) defines a homogeneous space provided ρ ( G ) is a transitive group of symmetries of the underlying set of X.

In a polynomial ring in some finite set of indeterminates, there is an action by ring automorphisms of the symmetric group on ( the indices of ) the indeterminates ( simultaneously substituting each of them for another according to the permutation used ).

For example, all order automorphisms of a power set S = 2 < sup > R </ sup > are induced by permutations of R. The papers cited above treat only sets S of functions on R < sup > n </ sup > satisfying some condition of convexity and prove that all order automorphisms are induced by linear or affine transformations of R < sup > n </ sup >.

Over an algebraically closed field, this is the only form ; however, over other fields, there are often many other forms, or “ twists ” of E < sub > 8 </ sub >, which are classified in the general framework of Galois cohomology ( over a perfect field k ) by the set H < sup > 1 </ sup >( k, Aut ( E < sub > 8 </ sub >)) which, because the Dynkin diagram of E < sub > 8 </ sub > ( see below ) has no automorphisms, coincides with H < sup > 1 </ sup >( k, E < sub > 8 </ sub >).

( The set of morphisms from the symmetric group S < sub > 3 </ sub > of order three to itself,, has ten elements: an element z whose product on either side with every element of E is z ( the homomorphism sending every element to the identity ), three elements such that their product on one fixed side is always itself ( the projections onto the three subgroups of order two ), and six automorphisms.

While in the first two cases the surface X admits infinitely many conformal automorphisms ( in fact, the conformal automorphism group is a complex Lie group of dimension three for a sphere and of dimension one for a torus ), a hyperbolic Riemann surface only admits a discrete set of automorphisms.

An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group.

Yet another equivalent definition of a rack is that it is a set where each element acts on the left and right as automorphisms of the rack, with the left action being the inverse of the right one.

The inner automorphisms of this group correspond to permuting the six elements from which we defined the morphemes and synthemes ; these permutations act on the Tutte – Coxeter graph by permuting the vertices on each side of its bipartition while keeping each of the two sides fixed as a set.

In the UNL approach, information conveyed by natural language is represented, sentence by sentence, as a hypergraph composed of a set of directed binary labeled links ( referred to as relations ) between nodes or hypernodes ( the Universal Words, or simply UW ), which stand for concepts.

Most classes of CSPs that are known to be tractable are those where the hypergraph of constraints has bounded treewidth ( and there are no restrictions on the set of constraint relations ), or where the constraints have arbitrary form but there exist essentially non-unary polymorphisms of the set of constraint relations.

Formally, a hypergraph is a pair where is a set of elements called nodes or vertices, and is a set of non-empty subsets of called hyperedges or edges.

A hypergraph is also called a set system or a family of sets drawn from the universal set X.

The difference between a set system and a hypergraph ( which is not well defined ) is in the questions being asked.

Given a subset of the index set, the partial hypergraph generated by is the hypergraph

A connected graph G with the same vertex set as a connected hypergraph H is a host graph for H if every hyperedge of H induces a connected subgraph in G. For a disconnected hypergraph H, G is a host graph if there is a bijection between the connected components of G and of H, such that each connected component G < nowiki >'</ nowiki > of G is a host of the corresponding H < nowiki >'</ nowiki >.

A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge.

A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices.

A transversal ( or " hitting set ") of a hypergraph H = ( X, E ) is a set that has nonempty intersection with every edge.

The transversal hypergraph of H is the hypergraph ( X, F ) whose edge set F consists of all minimal transversals of H.

The transpose of the incidence matrix defines a hypergraph called the dual of, where is an m-element set and is an n-element set of subsets of.

Consider, for example, the generalized hypergraph whose vertex set is and whose edges are and.

However, the transitive closure of set membership for such hypergraphs does induce a partial order, and " flattens " the hypergraph into a partially ordered set.

He set out to keep Troop H the best troop in the best regiment.

Throughout the 1970s and early 1980s, De Palma worked repeatedly with actors Jennifer Salt, Amy Irving, Nancy Allen ( his wife from 1979 to 1983 ), Gary Sinise, John Lithgow, William Finley, Charles Durning, Gerrit Graham, cinematographers Stephen H. Burum and Vilmos Zsigmond ( see List of noted film director and cinematographer collaborations ), set designer Jack Fisk, and composers Bernard Herrmann, John Williams and Pino Donaggio.

If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.

* the set of the elements of the new group is the cartesian product of the sets of elements of G and H, that is

His work and essays popularising the new understanding of radioactivity was the main inspiration for H. G. Wells's The World Set Free ( 1914 ), which features atomic bombs dropped from biplanes in a war set many years in the future.

This is a toilet of a brand new CityRail H set | OSCAR train, damaged in just within a period of 5 years

We define the kernel of h to be the set of elements in G which are mapped to the identity in H

On a recent anniversary of PGN an editorial in the Philadelphia Inquirer stated " Segal and PGN continue to step up admirably to the challenge set for newspapers by H. L.

H. 263 supports for a limited set of picture sizes:

H. 263v2 specified a set of recommended modes in an informative appendix ( Appendix II, since deprecated ):

where A is the arithmetic mean, H is the harmonic mean, M is the maximum of the interval and s < sup > 2 </ sup > is the variance of the set.

Figure 1: Complete acetylene ( H – C ≡ C – H ) molecular orbital set

* CityRail H set or Outer Suburban CARs ( OSCARs ), a type of train in Australia

IUPAC has proposed the use of a set of buffer solutions of known H < sup >+</ sup > activity.

INPUT: a set S of FDs that contain only subsets of a header H

Given an element φ of H *, the orthogonal complement of the kernel of φ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set.

* Steampunk is based on the idea of futuristic technology existing in the past, usually the 19th century, and often set in Victorian era England — but with prominent elements of either science fiction or fantasy, such as fictional technological inventions like those found in the works of H. G. Wells and Jules Verne, or real technological developments like the computer occurring at an earlier date.

It was not until 1972 that Magnavox released the first home video game console which could be connected to a TV set — the Magnavox Odyssey, invented by Ralph H. Baer.

Prinze's life and death were a focal point of one of the storylines in the movie Fame set in Prinze's alma mater Fiorello H. LaGuardia High School of Music & Art and Performing Arts.

If H is a subgroup of G, the set of left or right cosets G / H is a topological space when given the quotient topology ( the finest topology on G / H which makes the natural projection q: G → G / H continuous ).