Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

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.

Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.

An associative R-algebra is an additive abelian group A which has the structure of both a ring and an R-module in such a way that ring multiplication is R-bilinear:

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 ).

The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements.

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:

More compactly, an abelian group is a commutative group.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often ( but not always ) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.

The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian.

So in some sense it provides a measure of how far the group is from being abelian ; the larger the commutator subgroup is, the " less abelian " the group is.

In spite of the dreams of the host for oneness in the group, the Af incoming messages for each guest overload his receiving system beyond comprehension if N exceeds about six.

The poems which were addressed to her, while they are far more restrained than those of `` Love In Dian's Lap '', show no great technical advance over those of the `` Narrow Vessel '' group and are, if anything, somewhat more labored.

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.

Then, too, the utmost clinical flexibility is necessary in judiciously combining carefully timed family-oriented home visits, single and group office interviews, and appropriate telephone follow-up calls, if the worker is to be genuinely accessible and if the predicted unhealthy outcome is to be actually averted in accordance with the principles of preventive intervention.

For if the small group notion involves the implicit claim that the phenomena of sociological investigations are of atomic or subatomic proportions, the philosopher needs to know the extent to which such entities are valid.

The killer, if in our present group, would certainly be interested in knowing that much, and even though with the fingerprint evidence what it was I could see no way he could use this bit of information to improve on his situation, there might always be some way.

He said he would not be surprised if some of the more than 30 members of the group are interested in running on the required non-partisan ballot for posts on the charter commission.

It has controversially been argued by some evolutionary scientists such as E. O. Wilson that natural selection can act at the level of non-kin groups to produce adaptations that benefit a non-kin group even if these adaptions are detrimental at the individual level.

The leader is only a spokesperson for the group when it has to deal with other groups (" international relations ") but has no inside authority, and may be violently removed if he attempts to abuse this position.

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.

The percentage of fluent speakers turns out to be even higher if those under 16 are also taken into account, given that the proportion of bilinguals is particularly high in this age group ( 76. 7 % of those aged between 10 and 14 and 72. 4 % of those aged 5 – 9 ): 37. 5 % of the population aged 6 and above in the whole Basque Autonomous Community, 25. 0 % in Álava, 31. 3 % in Biscay and 53. 3 % in Gipuzkoa.

Another group under Sir John Simon then emerged, who were prepared to continue their support for the government and take the Liberal places in the Cabinet if there were resignations.

The professed goal of the group was the creation of a Greater Serbia, by use of violence, if necessary.

The group perfected its initial maneuvers in secret over the Florida Everglades so that, in Voris ' words, "... if anything happened, just the alligators would know.

In evolutionary biology, a group of organisms share common descent if they have a common ancestor.

If the carbonyl group is an aldehyde, the monosaccharide is an aldose ; if the carbonyl group is a ketone, the monosaccharide is a ketose.

The assignment of D or L is made according to the orientation of the asymmetric carbon furthest from the carbonyl group: in a standard Fischer projection if the hydroxyl group is on the right the molecule is a D sugar, otherwise it is an L sugar.