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.

Thus, if every historian were to claim that there was a solar eclipse in the year 1600, then though we might at first naively regard that as in violation of natural laws, we'd come to accept it as a fact.

Thus, a gene for antibiotic resistance that evolves via natural selection may be shared.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

Thus, any error in technology or judgment may have far-reaching consequences ( though the same can also happen through natural reproduction ).

Thus natural languages are mainly oral, while Blissymbols is just a writing system dealing with semantics, not phonetics.

Thus, what started as an effort to translate between languages evolved into an entire discipline devoted to understanding how to represent and process natural languages using computers.

Thus, to be valid, any law must conform to natural law and coercing people to conform to that law is morally acceptable.

Thus there is a natural bijection from the set of all possible equivalence relations on X and the set of all partitions of X.

Thus Esperanto achieved a stability of structure and grammar similar to that which natural languages enjoy by virtue of their native speakers and established bodies of literature.

Thus non-standard language varieties ( such as African American Vernacular English ) are considered to be natural while standard language varieties ( such as Standard American English ) which are more prescribed can be considered to be at least somewhat artificial or constructed.

Thus punctuated equilibrium contradicts some of Darwin's ideas regarding the specific mechanisms of evolution, but generally accords with Darwin's theory of evolution by natural selection.

Thus, a natural use of the turbocharger is with aircraft engines.

Thus, the moral value of one-celled organisms, as well as some multi-cellular organisms, and natural entities like a river, is only in the benefit they provide to sentient beings.

Thus, the effect of the accidental has to be understood in relation to the " natural " meaning of the note's staff position.

Thus, if in this example the composer actually wanted the note a semitone lower than G-natural, he might put first a natural sign to cancel the previous G-sharp, then the flat.

Thus a natural transformation is an infranatural transformation for which for every morphism.

Thus specific tissue effects of natural menopause cannot be attributed to loss of androgenic hormone production.

Thus T is uncountable: it cannot be placed in one-to-one correspondence with the set of natural numbers.

Thus, in a sense, the notion of a Noetherian ring unifies the ideal structure of various " natural rings ".

Thus no formal system ( satisfying the hypotheses of the theorem ) that aims to characterize the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove.

Thus, they are natural guardians of treasures or sacred sites which cannot easily be moved out of harm's way.

Thus an event will not constitute a natural disaster if it occurs in an area without vulnerability, e. g. an earthquake in an uninhabited area .< ref > Luis Flores Ballesteros.

Thus he won her fairly and took her away to Troia, with the full consent of her natural protectors.

Thus, for high-energy neutrons beryllium is a neutron multiplier, releasing more neutrons than it absorbs.

Thus lithium shows similarities to magnesium and beryllium to aluminum in many of their properties.

Thus, if what is at issue is whether `` All S is P '', it is indifferent whether `` Some S is not P '' or `` No S is P '', since in either case the judgment in question is false.

Thus, in any use of biological weapons, it is highly likely that animals will become ill either simultaneously with, or perhaps earlier than humans.

Thus, the Bayesian statistician needs either to use informed priors ( using relevant expertise or previous data ) or to choose among the competing methods for constructing " objective " priors.

Thus, a Britannica article either has known authorship

Thus, when either field is changing in time, then a field of the other is necessarily induced.

" Thus in the example above, Hecuba presents herself as a sophisticated intellectual describing a rationalized cosmos yet the speech is ill-matched to her audience, Menelaus ( a type of the unsophisticated listener ), and soon it is found not to suit the cosmos either ( her infant grandson is brutally murdered by the victorious Greeks ).

Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function.

Thus, the term " maternal surname " means the patrilineal surname which one's mother inherited from either or both of her parents.

Thus, in the current protocol, the queries carry the IP address and port number of either node.

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.

Thus we must write either to represent the former, or to represent the latter.

Thus (" The Father of Greatness ", the highest Manichaean deity of Light ), in Middle Persian texts might either be translated literally as pīd ī wuzurgīh, or substituted with the name of the deity Zurwān.

Thus the Masurian population in 1890, 143, 397 were reported by German officials as having German as their language ( either primary or secondary ), 152, 186 Polish, and 94, 961 Masurian.

Thus, a set of input objects is gathered and corresponding outputs are also gathered, either from human experts or from measurements.

Thus to Brouwer, we are not justified in asserting " either Goldbach's conjecture is true, or it is not.

Thus, a player who bids three and takes either two or four tricks will lose one point.

Thus the expression " point P is incident with line l " is used instead of either " P is on l " or " l passes through P ".

Thus Reversi can start with either a parallel or diagonal layout of the first 4 pieces.

Thus a conservative classification of Sino-Tibetan / Tibeto-Burman would posit several dozen small coordinate families and isolates ; attempts at subgrouping are either geographic conveniences or hypotheses for further research.

Thus, there are three major families of superoxide dismutase, depending on the metal cofactor: Cu / Zn ( which binds both copper and zinc ), Fe and Mn types ( which bind either iron or manganese ), and the Ni type, which binds nickel.

Thus, the hydroxide anion is represented by, the oxonium cation is and the cobalt III cation ( Co < sup > 3 +</ sup >) is either or.

Thus, in regard to this matter and this document, either one of two explanations is commonly held:

Thus, control unit X was either 1st or 5th, Y was either 2nd or 4th, and Z was 3rd in line.