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

It is, loosely speaking, the symmetry group of the object.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

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 automorphism group of an object X in a category C is denoted Aut < sub > C </ sub >( X ), or simply Aut ( X ) if the category is clear from context.

Each object represents a single entity — whether a user, a computer, a printer, or a group — and its attributes.

Most collective nouns encountered in everyday speech, such as " group ", are mundane and are not specific to one kind of constituent object.

Just as medieval group litigation bound all members of the group regardless of whether they all actually appeared in court, the modern class action binds all members of the class, with the exception of those who appear and object.

Deprogramming is commissioned by relatives, often parents of adult offspring, who object to someone's membership in an organization or group.

< font color ="# 0000ff "> global function </ font > delete_item ( < font color ="# ff00ff "> object </ font > old, < font color ="# ff00ff "> sequence </ font > group )

< font color ="# 0000ff "> global function </ font > replace_item ( < font color ="# ff00ff "> object </ font > old, < font color ="# ff00ff "> object </ font > new, < font color ="# ff00ff "> sequence </ font > group )

Some libertarians in the U. S. object to vouchers on the grounds that granting government money, even indirectly, to private and religious schools will inevitably lead to increased governmental control over non-government education, and possibly over the teachings of the sponsoring religious group ( most often a church ).

Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.

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.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

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.

More formally, a " representation " means a homomorphism from the group to the automorphism group of an object.

Compound comparisons typically compare two sets of groups means where one set has two or more groups ( e. g., compare average group means of group A, B and C with group D ).

Though not well known among philosophers, his philosophical work was taken up by Owen Barfield ( and through him influenced the Inklings, an Oxford group of Christian writers that included J. R. R. Tolkien and C. S. Lewis ) and Richard Tarnas.

* ISO C working group

Within this sonata form, the first group of the exposition starts out with a fugue before modulating to C major for the second part of the exposition.

**** C. medius group

**** C. major group

In April 2006, the Chadian rebel group United Front for Democratic Change, which is based in Darfur used C. A. R.

in negotiating a peace agreement was signed in Tripoli in February 2007, between President Bozizé and the head of the Front démocratique du people centrafricain ( FDPC ) rebel movement ( who is also said to have close ties to Union of Democratic Forces for Unity ( UFDR ), the rebel group that seized several cities in northern C. A. R.

In IUPAC nomenclature, organic compounds that have a – C ≡ N functional group are called nitriles.

The name may also mean an ester of carbonic acid, an organic compound containing the carbonate group C (= O )( O –)< sub > 2 </ sub >.

A typical monosaccharide has the structure H -( CHOH )< sub > x </ sub >( C = O )-( CHOH )< sub > y </ sub >- H, that is, an aldehyde or ketone with many hydroxyl groups added, usually one on each carbon atom that is not part of the aldehyde or ketone functional group.

The open-chain form of a monosaccharide often coexists with a closed ring form where the aldehyde / ketone carbonyl group carbon ( C = O ) and hydroxyl group (- OH ) react forming a hemiacetal with a new C-O-C bridge.

Thus, the molecular formula of ethanol may be written in a paper in conformational, three-dimensional, full two-dimensional ( indicating every bond with no three-dimensional directions ), compressed two-dimensional ( CH < sub > 3 </ sub >– CH < sub > 2 </ sub >– OH ), separating the functional group from another part of the molecule ( C < sub > 2 </ sub > H < sub > 5 </ sub > OH ), or by its atomic constituents ( C < sub > 2 </ sub > H < sub > 6 </ sub > O ), according to what is discussed.

A carboxyl group ( or carboxy ) is a functional group consisting of a carbonyl ( RR ' C = O ) and a hydroxyl ( R-O-H ), which has the formula-C (= O ) OH, usually written as-COOH or-CO < sub > 2 </ sub > H.

* Rhapsodies in Black: Art of the Harlem Renaissance, group show, 1998, Corcoran Gallery of Art, Washington, D. C.

A group is said to be of component type if for some centralizer C of an involution, C / O ( C ) has a component ( where O ( C ) is the core of C, the maximal normal subgroup of odd order ).

The content is not the same, however: rather than individual security, it is the security and continuing existence of an `` ideological group '' -- those in the `` free world '' -- that is basic.

Historically, however, the concept is one that has been of marked benefit to the people of the Western civilizational group.

The music director of the Pittsburgh Symphony Orchestra, William Steinberg, has molded his group into a prominent musical organization, which is his life.

With the group of boys it is different.

Nothing is more revealing of the way of life and literary aspirations of this group than their attitude toward sex.

This group is secularist and their program tends to be technological.

But the problem is one which gives us the measure of a man, rather than a group of men, whether a group of doctors, a group of party members assembled at a dinner to give their opinion, or the masses of the voters.

The group conducting the review is not holding formal hearings.

After allowing for group exposures, it is apparent that other factors must be considered if we are to comprehend fanaticism.

At Sounion there is a group of beautiful columns, the ruins of a temple to Poseidon, of particular interest at that time, as active reconstruction was in progress.

He is uncompromising in assigning guilt to the man who finds it necessary to inflict or permit injury to one individual or group for the sake of a larger good.

In the main stream of historical thinking is a group of scholars, H.M. Chadwick, R.H. Hodgkin, Sir Frank Stenton et al. who are in varying degrees sceptical of the native traditions of the conquest but who defend the catastrophic type of invasion suggested by them.

Although Patchen has given previous evidence of an interest in jazz, the musical group that he works with, the Chamber Jazz Sextet, is often ignored by jazz critics.

Patchen's musicians are outsiders in established jazz circles, and Patchen himself has remained outside the San Francisco poetry group, maintaining a self-imposed isolation, even though his conversion to poetry-and-jazz is not as extreme or as sudden as it may first appear.

Gov. Dalton's New Commerce and Industry Commission is moving to create a nine-state regional group in a collective effort to attract new industry.

Fortunately, there is a nursery school which he has been able to attend, with a group of normal children.

Perhaps Khrushchev is in a more difficult position than any since 1957, when the `` anti-party group '' nearly liquidated him.

When the fate of the individual is visited on the group, then ( the warm sweet butter dripped from her raised trembling fork and she pushed her head forward belligerently ), ah, then the true bitterness of existence could be tasted.

Mr. Claude is a specialist in torso development and he has long favored the now-famous Weider Push-Pull Super-Set technique in which one exercise of the Super-Set is a pressing or `` pushing '' movement which accents one sector of a muscle group in a specific way, followed by a `` pulling '' exercise which works the opposing sector of the same muscle group.