** For every non-empty set S there is a binary operation defined on S that makes it a group.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

For each element a of a group G, conjugation by a is the operation φ < sub > a </ sub >: G → G given by ( or a < sup >− 1 </ sup > ga ; usage varies ).

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.

To qualify as an abelian group, the set and operation,, must satisfy five requirements known as the abelian group axioms:

A group in which the group operation is not commutative is called a " non-abelian group " or " non-commutative group ".

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.

Continuing violence by British troops escalated, and eventually the order was given to mobilise the troops in an arrest operation, chasing the tail of the main group of marchers to the edge of the field by Free Derry Corner.

Often, one group of numbers in the instruction, called the opcode, indicates which operation to perform.

On September 28, 1995 Bob Denard and a group of mercenaries took over the Comoros islands in a coup ( named operation Kaskari by the mercenaries ) against President Djohar.

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

The subgroup of generated by all commutators is called the derived group or the commutator subgroup of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation.

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

From then on, in keeping with the traditions they had followed since childhood, the whole group settled down to relish their food.

Rod shifted his eager eyes from the milling group out in the circle long enough to reply, `` I ain't much of a hand for Dare-Base and Farmer-in-the-Dell, but I'd sure like to get in on the handhold and wrestles ''.

A few yards beyond the group of men, a man's nude body lay face down on a patch of thick green dichondra.

They arise in situations in which one believes that what happens depends not only on the external world, but also on the precise pattern of behavior of the individual or group.

Our collective policies, group and national, are similarly based on voodoo, but here we often lack even the empirically successful rituals and are still engaged in determing them.

In the following year her father undertook to give a course in Hebrew theology to Johns Hopkins students, and this brought to the Szold house a group of bright young Jews who had come to Baltimore to study, and who enjoyed being fed and mothered by Mamma and entertained by Henrietta and Rachel, who played and sang for them in the upstairs sitting room on Sunday evenings.

The group, upon the issuance of its first press release on December 21, 1957, designated itself a `` Committee of Investigation ''.

It sounded like a fair enough invitation, Peter Marshall reflected, and Bang-Jensen must have thought so too, because on the thirteenth, he met the group of three on the thirty-sixth floor of the U.N..

Peter Marshall noted that Bang-Jensen had later referred to his two interviews with the Gross group as `` unfortunate experiences '', and after his second meeting on the sixteenth the Dane refused to attend further hearings without legal counsel.

When the family business failed, Mercer left school and on his mother's urging -- for she hoped that he would become an actor -- he joined a local little theater group.

Like his volume on Wycliffe, the work was accompanied by the publication of a selected group of documents, in this case illustrative of the history of Queen Anne's reign down to 1707.

Such a situation regarding the Board of War could hardly have helped Morgan's chances for promotion when that matter came before the group later on.

The board of suspension of the Interstate Commerce commission has ordered a group of railroads not to reduce their freight rates on grain, as they had planned to do this month.

`` If the day should ever come that foreign invaders swarm ashore along the Gulf Coast '', the account reads, `` they can count on heavy opposition from a group of commando-trained telephone employees -- all girls.

One cannot but wonder whether these doubts about the success of Khrushchev's agricultural policy have not at least something to do with one of the big surprises provided by this Congress -- the obsessive harping on the crimes and misdeeds of the `` anti-party group '' -- Molotov, Malenkov, Kaganovich and others -- including the eighty-year-old Marshal Voroshilov.

But one cannot escape the suspicion that all this non-stop harping on the misdeeds of the long liquidated `` anti-party '' group would be totally unnecessary if there were not, inside the party, some secret but genuine opposition to Khrushchev on vital doctrinal grounds, on the actual methods to be employed in the `` transition to communism '' and, last but not least, on foreign policy.

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.

Centering around this historic old structure, a group of public-spirited Barbour County citizens have organized and planned a week-long series of events, beginning on May 28th and continuing through June 3rd, to observe most appropriately the centennial of the first land engagement of the Civil War at Philippi.

Are annual marketing plans reviewed throughout your management group to get the perspective of all individuals and get everyone on the marketing team??

The largest of these organizations at present denies to the full time educator any vote on the conduct and standards of the group and, indeed, refuses him even the right to attach the customary initials after his name in the college catalog.

In 1428, he received a letter from Yeshaq I of Ethiopia, borne by two dignitaries, which proposed an alliance against the Muslims and would be sealed by a dual marriage that would require the Infante Peter to bring a group of artisans to Ethiopia where he would marry Yeshaq's daughter.

The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one.

There is a notion of ind-finite group, which is the concept dual to profinite groups ; i. e. a group G is ind-finite if it is the direct limit of an inductive system of finite groups.

In current scholarship, the standard assumption is to group the Ionic together with the Arcadocypriot group as the successors of a single Middle Bronze Age migration in dual opposition to the " western " group of Doric.

The 3-adic integers, with selected corresponding characters on their Pontryagin dual group

Ieuan Wyn Jones, who had resigned from his dual role as president and Assembly group leader following the losses in the 2003 Assembly election, was re-elected in the latter role.

* Given a category C with finite coproducts, a cogroup object is an object G of C together with a " comultiplication " m: G → G G, a " coidentity " e: G → 0, and a " coinversion " inv: G → G, which satisfy the dual versions of the axioms for group objects.

Furthermore, given such a group G, Langlands constructs the Langlands dual group < sup > L </ sup > G, and then, for every automorphic cuspidal representation of G and every finite-dimensional representation of < sup > L </ sup > G, he defines an L-function.

An energetic set of demos ( eight of which were briefly and illegally released on Pickwick Records in 1980 under the title Introducing the Doobie Brothers, and have since been bootlegged on CD under that title and On Our Way Up as well, both with expanded song selections ), showcased fuzz-toned dual lead electric guitars, three-part harmonies and Hartman's frenetic drumming and earned the rock group a contract at Warner Bros. Records.

All persons holding a dual hackney carriage or private hire driver license within the City of Chelmsford must meet strict criteria as laid down by the authority which includes license renewal every 2 years, a Criminal Records Bureau enhanced disclosure check every three years and a Driver and Vehicle Licensing Agency group 2 medical examination every four years.

With their redesignated dual missions, the Vega probes became part of the Halley Armada, a group of space probes that studied Halley's Comet during its 1985 / 86 perihelion.

Queen Elizabeth II is dead, killed in an arbitrary mugging, and Elizabeth I moves through the social and physical decay of the city observing the activities of a group of sporadic nihilists, including Amyl Nitrite ( Jordan ), Bod ( Runacre in a dual role ), Chaos ( Hermine Demoriane ), Crabs ( Nell Campbell ), and Mad ( Toyah Willcox ).

A small group of students also choose to apply for a dual degree, allowing them to graduate with two degrees — a B. S.

The dances themselves are mainly group dances rather than individual or dual dances.

When one wants to work over a base ring R ( commutative ), there is the group scheme concept: that is, a group object in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra.

This group is self dual in the sense that it is topologically isomorphic to its group of characters.

There is a dual theory, group homology, and a generalization to non-abelian coefficients.