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.

For example, in peas the allele for green pods, G, is dominant to that for yellow pods, g. Thus pea plants with the pair of alleles either GG ( homozygote ) or Gg ( heterozygote ) will have green pods.

This identifies the tangent space T < sub > e </ sub > at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur Thus the Lie bracket on is given explicitly by = < sub > e </ sub >.

Thus for a theory " T ", " G " is true, but not provable in " T ".

Thus, in DNA, the purines adenine ( A ) and guanine ( G ) pair up with the pyrimidines thymine ( T ) and cytosine ( C ), respectively.

Thus, as Time magazine wrote during the controversial tenure of Vice President Charles G. Dawes, " once in four years the Vice President can make a little speech, and then he is done.

Thus one can speak unambiguously of the decidability of the word problem for the finitely generated group G.

Thus a D whistle is fairly apt for playing both G and A, and a C instrument can be used fairly easily for F and G.

Thus, G < sub > t </ sub >( V ) = exp ( tV ) is the exponential map of the vector tV.

Thus, the G. 711 encoder will create a 64 kbit / s bitstream for a signal sampled at 8 kHz.

Thus, f < sub > g </ sub > is a permutation of G, and so is a member of Sym ( G ).

Thus G is isomorphic to the image of T, which is the subgroup K.

In one direction, the Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower ( in the worst case, as a function of the number of vertices ) than finding a Hamiltonian cycle.

Thus if G is any finitely generated group of exponent n, then G a homomorphic image of B ( m, n ), where m is the number of generators of G. Burnside's problem can now be restated as follows:

Thus, to cite but one example, the Pax Britannica of the nineteenth century, whether with the British navy ruling the seas or with the City of London ruling world finance, was strictly national in motivation, however much other nations ( e.g., the United States ) may have incidentally benefited.

Thus we are compelled to face the urbanization of the South -- an urbanization which, despite its dramatic and overwhelming effects upon the Southern culture, has been utterly ignored by the bulk of Southern writers.

Thus Faulkner reminds us, and wisely, that the `` new '' South has gradually evolved out of the Old South, and consequently its agrarian roots persist.

Thus, the avant-garde choreographers have extended the scope of materials available for dance composition.

Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.

Thus, paradoxically, the beat writers resort to `` religious '' metaphors: they are in search of mana, the spiritual, the numinous, but not anything connected with formal religion.

Thus jazz is transmuted into something holy, the sacred road to integration of being.

Thus, on the highroad, a troop of soldiers `` marched in their own dust and sang '', while on the footpath one man walks alone.

Thus the image of man has suffered complete fragmentation in personal and spiritual qualities, and complete objectification in sub-human and quasi-mechanistic powers.

Thus the fictional detective is much more than a simple businessman.

Thus the transformation of Adam Smith's ideal entrepreneur into a mythological detective coincides closely with the decline of the real entrepreneur in economic life.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Thus, if corporations are not to run away with us, they must become quasi-governmental institutions, subject to public control and needs.

Thus in both types attention is focused on the community itself, and its phenomenological life.

Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.

Thus human perception and human volition is the immanent cause of all social change and this most truly when the change reaches the civilizational level.

Thus with regard to the loss of tradition, in the change from order to disorder the metaphysics of change works itself out as a disruption of the individual soul, a change in which man continues as an objective ontological existent, but no longer as a man.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus earth has fallen to the center of the universe.

Thus, Margenau remarks: `` A large number of unrelated epicycles was needed to explain the observations, but otherwise the ( Ptolemaic ) system served well and with quantitative precision.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

Thus he complains, with considerable justice, that the Tory writers have resorted to libel instead of answering his arguments.

Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.

Thus the student of literature may sometimes find it helpful to classify a poem or an essay as being in idea or in ideal content or subject matter typical or atypical of its period.

Thus ideas like `` grace '', `` salvation '', and `` providence '' cluster together in traditional Christianity.

If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Thus if the universe is hot enough ( approximately 10 < sup > 15 </ sup > K, a temperature exceeded until shortly after the Big Bang ) then the electromagnetic force and weak force will merge into a combined electroweak force.

Thus K < sub > V </ sub > 1. 3 blockers could be potential immunosuppressants for the treatment of autoimmune disorders ( such as rheumatoid arthritis, inflammatory bowel disease and multiple sclerosis ).

Thus for a transition, K represents the core level hole, the relaxing electron's initial state, and the emitted electron's initial energy state.

Così fan tutte, ossia La scuola degli amanti ( Thus Do They All, or The School For Lovers ) K. 588, is an Italian language opera buffa by Wolfgang Amadeus Mozart first performed in 1790.

Thus at 100 K the energy flux density is 5. 67 W / m < sup > 2 </ sup >, at 1000 K 56, 700 W / m < sup > 2 </ sup >, etc.

Thus, for calcium sulfate, K < sub > sp </ sub > =, log K < sub > sp </ sub > =-4. 32.

Thus the queen's file is named " Q " and the king's file is named " K ".

Thus in July 1981 " VME2900 " was launched: although presented to the customer base as a merger of VME / B and VME / K, it was in reality the VME / B base with a few selected features from VME / K grafted on.

1, points J, K, and L .) Thus, Terquem was the first to use the name nine-point circle.

Thus, the area K of a triangle may be found by multiplying the inradius by the semiperimeter:

Thus the ' classic ' Uriah Heep was formed and, according to biographer K. Blows, " everything just clicked into place ".

Thus, substances causing an increase in cAMP upregulate the Na < sup >+</ sup >/ K < sup >+</ sup >- ATPase.

Thus, given the field K, any algebra can be specified up to isomorphism by giving its dimension ( say n ), and specifying n < sup > 3 </ sup > structure coefficients c < sub > i, j, k </ sub >, which are scalars.

Note that the right-hand side of the equation is also the price of buying a forward contract on the stock with delivery price K. Thus one way to read the equation is that a portfolio that is long a call and short a put is the same as being long a forward.

Thus, its divisor K :=

Thus, b < sub > q </ sub > is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form

Thus, replacing E by K we may assume that f < sub > n </ sub > converge to f pointwise on E. Next, by the definition of the Lebesgue Integral, it is enough to show that if φ is any non-negative simple function less than or equal to f, then