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.

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

Since then, and since the pure grain had gotten him divorced from every decent -- and even indecent -- group from Greenwich Village to the Embarcadero, he had become a sucker-rolling freight-jumper.

And then the Amen corner took hold, re-enacting a form of group participation in worship that stemmed from years before the Greek chorus, spreading down through the African forest, overseas to the West Indies, and then here in Alabama.

We quickly ran into the same trouble that plagued Bill Ruger in his first experiments: Three or four bullets would be placed well in a six-inch bull at 100 yards and then, unaccountably, one could stray far out of the group.

At the very first, then, Brumidi was required, by the classically pyramidal shape of his central group, to fill in the triangular space above the seated girl on Liberty's right, before starting on the allegorical figures themselves.

She got up, standing uncertainly for a moment, then moving aside to let go past her, talking, a group of young men.

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 oxidation pathway starts with the removal of the amino group by a transaminase, the amino group is then fed into the urea cycle.

When Troy was sacked by the Greeks, Aeneas, after being commanded by the gods to flee, gathered a group, collectively known as the Aeneads, who then traveled to Italy and became progenitors of Romans.

In 1992 a feminist group began a nationwide fundraising campaign and then obtained matching state funds to endow a professorship at the University of Oklahoma Law School in honor of Hill.

In mid-November 2000, the 9 / 11 Commission believes that Nami, Wail and Waleed al-Shehri, all of whom had obtained their U. S. visas in late October, traveled in a group from Saudi Arabia to Beirut and then onward to Iran where they could travel through to Afghanistan without getting their passports stamped.

Two monosaccharides can be joined together using dehydration synthesis, in which a hydrogen atom is removed from the end of one molecule and a hydroxyl group (— OH ) is removed from the other ; the remaining residues are then attached at the sites from which the atoms were removed.

This is important in the biosynthesis of amino acids, as for many of the pathways, intermediates from other biochemical pathways are converted to the α-keto acid skeleton, and then an amino group is added, often via transamination.

The Roman empire under Hadrian ( ruled 117-38 ), showing the location of the Burgundiones Germanic group, then inhabiting the region between the Viadua ( Oder ) and Visula ( Vistula ) rivers ( Poland )

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.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

A group of bishops and divines met first at Chertsey and then at Windsor in 1548, drawn from both conservatives and reformers, agreed only " the service of the church ought to be in the mother tongue ".

Bozizé then fled to Chad and gathered a group of rebellions.

The cerebral cortex is a group of tight, dense, " gray matter " composed of the nucleus of the neurons whose axons then form the " white matter ", and is responsible for perception, relay of the sensory input ( sensation ) via the thalamic pathway, and most importantly directly or indirectly in charge of all the neurological functions, from simple reflexes to complex thinking.

The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms.