It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is diffeomorphic to the circle.

It has a closed connected subgroup SL < sub > n </ sub >( R ), the special linear group, consisting of matrices of determinant 1 which is also a Lie group.

* The unitary group U ( n ) consisting of n × n unitary matrices ( with complex entries ) is a compact connected Lie group of dimension n < sup > 2 </ sup >.

It is a connected Lie group of dimension 2n < sup > 2 </ sup > + n.

* The Heisenberg group is a connected nilpotent Lie group of dimension 3, playing a key role in quantum mechanics.

This is a one-dimensional compact connected abelian Lie group.

It is a connected Lie group that cannot be faithfully represented by matrices of finite size, i. e., a nonlinear group.

* The universal cover of a connected Lie group is a Lie group.

Lie groups are classified according to their algebraic properties ( simple, semisimple, solvable, nilpotent, abelian ), their connectedness ( connected or simply connected ) and their compactness.

* Compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S < sup > 1 </ sup > and simple compact Lie groups ( which correspond to connected Dynkin diagrams ).

* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

* Simple Lie groups are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra.

The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center.

The Charles Men consists not of a connected narrative but of a group of short stories, each depicting a special phase of the general subject.

A canid race, each ' person ' comprising a group mind of 5-8 individuals, connected by sound-waves.

This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type.

Ethers () are a class of organic compounds that contain an ether group — an oxygen atom connected to two alkyl or aryl groups — of general formula R – O – R '.

A path-connected space with a trivial fundamental group is said to be simply connected.

If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

Representation of an organic compound | organic hydroxyl group, where R represents a hydrocarbon or other organic moiety, the red and grey spheres represent oxygen and hydrogen atoms, respectively, and the rod-like connections between these, covalent chemical bond s. A hydroxyl is a chemical functional group containing an oxygen atom connected by a covalent bond to a hydrogen atom, a pairing that can be simply understood as a substructure of the water molecule.

The scandal began as an operation to free seven American hostages being held by a group with Iranian ties connected to the Army of the Guardians of the Islamic Revolution.

* In homotopy theory, the fundamental group of a space at a point, though technically denoted to emphasize the dependence on the base point, is often written lazily as simply if is path connected.

Unlike the four larger Scottish archipelagos, none of the isles in this group are connected to one another or to the mainland by bridges.

Faderman calls this period " the last breath of innocence " before 1920 when characterizations of female affection were connected to sexuality, marking lesbians as a unique and often unflattering group.

This group is disconnected ; it has two connected components corresponding to the positive and negative values of the determinant.

This group is disconnected, but it has a connected subgroup SO < sub > n </ sub >( R ) of the same dimension consisting of orthogonal matrices of determinant 1, called the special orthogonal group ( for n = 3, the rotation group SO ( 3 )).

His religiousness is intimately, or dialectically, connected with his sinfulness ; ;

The fact that sticks out in this voluminous record is that the bulk of Du Pont's production has always supplied the largest part of the requirements of the one customer in the automobile industry connected to Du Pont by a stock interest.

The main set of bars are the `` tappets '' and one tappet is connected to each lever.

Only the independent art schools, that is, those not connected with any university or college, receive severe and separate investigation before accreditation by the various regional organizations.

But Mother insisted, for it is seldom indeed that anyone remotely connected with the cinema is ever received in their exclusive midsts.

It is connected by teletype with the State Library in Albany, which will supply any book to a system that the system itself cannot provide.

The function of Apollo as a " healer " is connected with Paean ( Παιών-Παιήων ), the physician of the Gods in the Iliad, who seems to come from a more primitive religion.

Paeοn is probably connected with the Mycenean Pa-ja-wo, but the etymology is the only evidence.

The connection with Dorians and their initiation festival apellai is reinforced by the month Apellaios in northwest Greek calendars, but it can explain only the Doric type of the name, which is connected with the Ancient Macedonian word " pella " ( Pella ), stone.

A female dragon named Delphyne ( δελφύς: womb ), who is obviously connected with Delphi and Apollo Delphinios, and a male serpent Typhon ( τύφειν: smoke ), the adversary of Zeus in the Titanomachy, who the narrators confused with Python.

The old oracles in Delphi seem to be connected with a local tradition of the priesthood, and there is not clear evidence that a kind of inspiration-prophecy existed in the temple.

The earliest attested name is the Hittite Assuwa a region in central-western Anatolia which seems to be connected with the Mycenean Greek epithet a-si-wi-ja in Linear B inscriptions found at Pylos.

As one component of the interconnected global ocean, it is connected in the north to the Arctic Ocean ( which is sometimes considered a sea of the Atlantic ), to the Pacific Ocean in the southwest, the Indian Ocean in the southeast, and the Southern Ocean in the south.

In the north, it is connected to the Marmara Sea and Black Sea by the Dardanelles and Bosporus.

The main feature of the family is the composite flower type in the form of capitula surrounded by involucral bracts. The name " Asteraceae " comes from Aster, the most prominent generum in the family, that derives from the Greek ἀστήρ meaning star, and is connected with its inflorescence star form.

The legend connected with its foundation is given by Peter Damiani in his Life of St Odilo: a pilgrim returning from the Holy Land was cast by a storm on a desolate island.

Antisemitism ( also spelled anti-semitism or anti-Semitism ) is suspicion of, hatred toward, or discrimination against Jews for reasons connected to their Jewish heritage.

Furthermore, the primary testimony to the commission that connected baseball to Doubleday was that of Abner Graves, whose credibility is questionable ; a few years later, he shot his wife to death and was committed to an institution for the criminally insane for the rest of his life.

The noun is rarely used in American English to refer to people not connected to the United States.

His work in this specific field ( based on the criss-crossing between literary criticism, bibliography, and sociocultural history ) is connected to broader historiographical and methodological interests which deal with the relation between history and other disciplines: philosophy, sociology, anthropology.