This was essentially solved by Aschbacher's trichotomy theorem for groups with e ( G )= 3.

Namely φ is universal for homomorphisms from G to an abelian group H: for any abelian group H and homomorphism of groups f: G → H there exists a unique homomorphism F: G < sup > ab </ sup > → H such that.

Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ ( ab < sup >− 1 </ sup > ∈ H ).

A study analyzed voting records of the Fifth European Parliament and ranked groups, concluding: " Towards the top of the figure are the more pro-European parties ( PES, EPP-ED, and ALDE ), whereas towards the bottom of the figure are the more anti-European parties ( EUL / NGL, G / EFA, UEN and EDD ).

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

In mathematics, given two groups ( G, *) and ( H, ·), a group homomorphism from ( G, *) to ( H, ·) is a function h: G → H such that for all u and v in G it holds that

Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.

The two groups (< var > G </ var >, *) and (< var > H </ var >, ) are isomorphic if there exists an isomorphism from one to the other.

Carboxylic acid groups (− CO < sub > 2 </ sub > H ) can be deprotonated to become negative carboxylates (− CO < sub > 2 </ sub >< sup >−</ sup > ), and α-amino groups ( NH < sub > 2 </ sub >−) can be protonated to become positive α-ammonium groups (< sup >+</ sup > NH < sub > 3 </ sub >−).

An amide is a compound with the functional group R < sub > n </ sub > E ( O )< sub > x </ sub > NR '< sub > 2 </ sub > ( R and R ' refer to H or organic groups ).

The simplest acyclic alkenes, with only one double bond and no other functional groups, form an homologous series of hydrocarbons with the general formula C < sub > n </ sub > H < sub > 2n </ sub >.

The two groups are Emanuel Maxwell, who published his results in Isotope Effect in the Superconductivity of Mercury and C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt who published their results 10 pages later in Superconductivity of Isotopes of Mercury.

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 A-series, B-series, C-series and D-series, whose elements are denoted by A < sub > n </ sub >, B < sub > n </ sub >, C < sub > n </ sub >, and D < sub > n </ sub >, are infinite families of simple Lie groups.

* the infinite discrete groups generated by a translation and a reflection in a point ; they are isomorphic with the generalized dihedral group of Z, Dih ( Z ), also denoted by D < sub >∞</ sub > ( which is a semidirect product of Z and C < sub > 2 </ sub >).

The etymology cannot be accurately determined, for historically the groups it denoted were not ethnically fixed, since it denoted a political rather than a tribal identity, or was used originally to refer to just one group among several, the others calling themselves Toquz Oghuz.

Phenyl groups are often represented by the symbol Ph or, archaically, Φ. Benzene is sometimes denoted as PhH.

The following table summarizes the various ethnic groups and / or other social groups of known historical significance to the history of China ( any non clear-cut connection is denoted by a question mark ):

The Scots-Irish settled in a dispersed community pattern denoted by communal, clannish, family-related groups known as " clachans ", much the same as in Pennsylvania and Ulster, Northern Ireland.

Cohomology groups are formally similar: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted d < sup > n </ sup > point in the direction of increasing n rather than decreasing n ; then the groups and follow from the same description and

The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups ( which are called cohomology groups in this context and denoted by H < sup > n </ sup >) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.

Given two groups A and H there exist two variations of the wreath product: the unrestricted wreath product A Wr H ( also written A ≀ H ) and the restricted wreath product A wr H. Given a set Ω with an H-action there exists a generalisation of the wreath product which is denoted by A Wr < sub > Ω </ sub > H or A wr < sub > Ω </ sub > H respectively.

) By analogy with GL / SL ( general linear group, special linear group ), the orthogonal group is sometimes called the general orthogonal group and denoted GO, though this term is also sometimes used for indefinite orthogonal groups O ( p, q ).

In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp ( 2n, F ) and Sp ( n ).

We may therefore form its right derived functors ; their values are abelian groups and they are denoted by H < sup > n </ sup >( G, M ), " the n-th cohomology group of G with coefficients in M ".

Some groups choose to host a single speaker ( such meetings are usually denoted " speaker meetings ") to share for the majority of the meeting time.

The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the categories of finite G-sets for a fixed profinite group G. For example, G might be the group denoted, which is the inverse limit of the cyclic additive groups Z / nZ — or equivalently the completion of the infinite cyclic group Z for the topology of subgroups of finite index.

To perpetuate wealth control led by small groups of individuals who played no role in its creation prevents those with real initiative from coming to the fore, and is basically anti-democratic.

It is a fact of life that magazines are edited by groups: they have to be or they wouldn't be published at all.

The arguments advanced by those individuals and groups who oppose the system in force and who would drastically curtail or do away entirely with hospital care for the non-service-connected case, seem to be coldly impractical and out-of-step with the wishes of the general public.

On the one hand, the Public Health Service declared as recently as October 26 that present radiation levels resulting from the Soviet shots `` do not warrant undue public concern '' or any action to limit the intake of radioactive substances by individuals or large population groups anywhere in the Aj.

There are other countries where, with skillful diplomacy, we may be able by our aid to give encouragement to those groups in government which would like to press forward with economic and social reform measures to promote growth.

In the same period, 431 presentations by members of the staff were made to local, national, and international medical groups.

Inventory and evaluate wildlife habitat resources in cooperation with other Federal agencies and with the States in which National Forests and Grasslands are located, as a basis for orderly development of wildlife habitat improvement and coordination programs, including ( A ) big-game, gamebird, and small-game habitat surveys and investigations on the 186 million acres of National Forests and Grasslands, ( B ) fishery habitat surveys and investigations on the 81,000 miles of National Forest fishing streams and nearly 3 million acres of lakes and impoundments, and ( C ) participation in planning, inspection, and control phases of all habitat improvement, land and water use projects conducted on National Forest lands by States, other Federal agencies, and private groups to assure that projects will benefit wildlife and be in harmony with other resource values.

Active groups sponsored by the organization include the Saddle Club, Orchesis, Golf Club, Tennis Club, and Dolphins.

Modifications of the last technique have been applied by several groups of investigators.

The most active preparations obtained by these two groups of investigators appear to be similar in potency, composition and physical properties.

Educational policies are formed by several groups who are officially or unofficially appointed to act in the public interest.

Though Americans usually lived in groups segregated by national origin or religious belief, they liked to work and shop in the noise and vitality of downtown.

( Mussorgsky cleverly contrasts the two groups by their orchestral accompaniment, solemn chords or mocking staccatos.

Part 1, deals with the classification of crystalline substances by space groups and is not a numerical data compilation.

Do patriots everywhere know enough about how the persecution of the Jews in Germany and later in the occupied countries contributed to terrorizing the populations, splitting apart individuals and groups, arousing the meanest and most dishonest impulses, pulverizing trust and personal dignity, and finally forcing people to follow their masters into the abyss by making them partners in unspeakable crimes??

Dominated by the vicious circle of the university promotion system, the analyst looks down on and gets along poorly with the other three groups, although he cannot deny his debt to the collector.

It appears that an Orthodox girl in the community disrupted plans for an outing sponsored by one of the Jewish service groups because she would not travel on Saturday and, in addition, required kosher food.

However, if the child has been constantly surrounded, during nursery and early school age, by peer groups ; ;

This group pleads with the administration to `` give no further support for the invasion of Cuba by exile groups ''.

`` I would expect the proposed committee to hold public hearings '', Mr. Notte said, `` to obtain the views of the general public and religious, labor and special-interest groups affected by these laws ''.

About half of all Peace Corps projects assigned to voluntary agencies will be carried out by religious groups, according to an official of the corps.

Twelve projects proposed by private groups are at the contract-negotiation stage, Gordon Boyce, director of relations with the voluntary agencies, said in a Washington interview.

Six of these were proposed by religious groups.

Sample copies of new books are on display at headquarters, where librarians may evaluate them by themselves or in workshop groups.

Shoals of tourists went by her in national groups -- English school girls in blue uniforms, German boys with cameras attached, smartly dressed Americans looking in shop windows.