The quotient group Aut ( G ) / Inn ( G ) is usually denoted by Out ( G ); the non-trivial elements are the cosets that contain the outer automorphisms.

One way in which the quotient set resembles division is that if is finite and the equivalence classes are all equinumerous, then the number of equivalence classes in can be calculated by dividing the number of elements in by the number of elements in each equivalence class.

In mathematics, specifically group theory, a quotient group ( or factor group ) is a group obtained by identifying together elements of a larger group using an equivalence relation.

Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.

Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes ( or congruence classes ) for the relation.

Together, these equivalence classes are the elements of a quotient group.

In yet a simpler way, we may think of the Jacobson radical of a ring as method to " mod out bad elements " of the ring – that is, members of the Jacobson radical act as 0 in the quotient ring, R / J ( R ).

If N is the nilradical of commutative ring R, then the quotient ring R / N has no nilpotent elements.

Consider the polynomial ring R, and the irreducible polynomial The quotient space is given by the congruence As a result, the elements ( or equivalence classes ) of are of the form where a and b belong to R. To see this, note that since it follows that,,, etc.

One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R / I, essentially by requiring that all elements of I be zero.

Intuitively, the quotient ring R / I is a " simplified version " of R where the elements of I are " ignored ".

This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over K are again algebraic over K. The set of all elements of L which are algebraic over K is a field that sits in between L and K.

The principal idèles are given by the diagonal embedding of the invertible elements of the number field or field of functions and the quotient of the idele group by principal ideles is the idele class group.

The Z < sub > 2 </ sub > in the quotient refers to the two element subgroup generated by the element of the center corresponding to the 2 element of Z < sub > 4 </ sub > and the 1 elements of Z < sub > 2L </ sub > and Z < sub > 2R </ sub >.

In mathematics, a quotient algebra, ( where algebra is used in the sense of universal algebra ), also called a factor algebra is obtained by partitioning the elements of an algebra in equivalence classes given by a congruence, that is an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below.

His major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n − 1.

Since X is not a group we cannot multiply elements ; we can, however, take their " quotient ".

See Weyl algebra for this ; one must take a quotient, so that the central elements of the Lie algebra act as prescribed scalars.

Arithmetic operators are,,, and ( binary operators, which pop two elements from the stack and push ( respectively ) their sum, difference, product, or quotient ) and the ( underscore ) is unary negation ( which pops one element and pushes its negation ).

: The quotient ring is the finite field with p < sup > 2 </ sup > elements: O < sub > K </ sub >/ pO < sub > K </ sub >

: The quotient ring contains non-zero nilpotent elements.

The norms, as elements, refer to `` all criteria for judging the character or conduct of both individual and group actions in any social system ''.

The adherence of many in the population to the Indian background in their pedigree, and emphasis upon the fact that their ancestors had never been slaves, becomes of prime interest in determining how far these elements promote the self-image of the intermediate status of the group in society.

Where boundary maintenance describes the boundaries or limits of the group, systemic linkage is defined `` as the process whereby one or more of the elements of at least two social systems is articulated in such a manner that the two systems in some ways and on some occasions may be viewed as a single unit.

Design elements closely rooted to traditional forms but wearing a definite contemporary label keynote Drexel's fall 1961 group, Composite.

Karma is categorized within the group or groups of cause ( Pāli hetu ) in the chain of cause and effect, where it comprises the elements of " volitional activities " ( Pali sankhara ) and " action " ( Pali bhava ).

The alkali metals are a group of chemical elements in the periodic table with very similar properties: they are all shiny, soft, silvery, highly reactive metals at standard temperature and pressure and readily lose their outermost electron to form cations with charge + 1.

In the modern IUPAC nomenclature, the alkali metals comprise the group 1 elements, excluding hydrogen ( H ), which is nominally a group 1 element but not normally considered to be an alkali metal as it rarely exhibits behaviour comparable to that of the alkali metals.

The alkali metals provide the best example of group trends in properties in the periodic table, with elements exhibiting well-characterized homologous behaviour.

Experiments have been conducted to attempt the synthesis of ununennium ( Uue ), which is likely to be the next member of the group, but they have all met with failure .< ref name =" link "> However, ununennium may not be an alkali metal due to relativistic effects, which are predicted to have a large influence on the chemical properties of superheavy elements.

Actinium gave the name to the actinide series, a group of 15 similar elements between actinium and lawrencium in the periodic table.

The group of elements is more diverse than the lanthanides and therefore it was not until 1945 that Glenn T. Seaborg proposed the most significant change to Mendeleev's periodic table, by introducing the actinides.

The separation of curium and americium was so painstaking that those elements were initially called by the Berkeley group as pandemonium ( from Greek for all demons or hell ) and delirium ( from Latin for madness ).

Informally, it is a permutation of the group elements such that the structure remains unchanged.

In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself.

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order ( the axiom of commutativity ).

An abelian group is a set, A, together with an operation "•" that combines any two elements a and b to form another element denoted.

* Base ( group theory ), a sequence of distinct elements of a set

In the upper Dniester, where the Basternae are described to have inhabited, post-Zarubintsy, Przeworsk and Sarmatian elements formed the Zvenigrod group, which also has some analogies with the " Dacian " Lipitsa culture.

The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous, ( and hence homeomorphism ) so that it forms a topological group under multiplication.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

That is, if R is a PID, and a and b are elements of R, and d is a greatest common divisor of a and b,

then there are elements x and y in R such that ax + by = d. The reason: the ideal Ra + Rb is principal and indeed is equal to Rd.

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are " a ~ b " and " a ≡ b ", which are used when R is the obvious relation being referenced, and variations of " a ~< sub > R </ sub > b ", " a ≡< sub > R </ sub > b ", or " aRb ".

This yields a convenient way of generating an equivalence relation: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub > in X such that a

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b ( that is, if there are elements x and y in R such that d · x = a and d · y = b ).

If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

However if R is a unique factorization domain, then any two elements have a gcd, and more generally this is true in gcd domains.

* Gal ( C / R ) has two elements, the identity automorphism and the complex conjugation automorphism.

If R is the direct sum of the rings R < sub > 1 </ sub >,..., R < sub > n </ sub >, then the identity elements of the rings R < sub > i </ sub > are central idempotents in R, pairwise orthogonal, and their sum is 1.