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If the inner automorphism group of a perfect group G is simple, then G is called quasisimple.
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If and inner
If the inner functions of religion are performed, the individual is a composed, ordered, motivated, and emotionally secure associate ; ;
If the gas is inhaled, some of the radon particles may attach to the inner lining of the lung.
If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.
Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.
If μ is both inner regular and locally finite, it is called a Radon measure.
If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.
The parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
If one alters a Euclidean space so that its inner product becomes negative in one or more directions, then the result is a pseudo-Euclidean space.
If a charged Leyden jar is discharged by shorting the inner and outer coatings and left to sit for a few minutes, the jar will recover some of its previous charge, and a second spark can be obtained from it.
If no nucleus is present, but irregular and small dark inner spots indicating a cavity are visible, combined with concentric rings of organic substance, the pearl is likely a cultured freshwater.
If and are any complex numbers, the inner product is the standard inner product and the bar notation is used for complex conjugation then the inequality may be restated in more explicitly as
If Halley was once a long-period comet, it is likely to have originated in the Oort Cloud, a sphere of cometary bodies that has an inner edge of 20, 000 – 50, 000 AU.
If Earth's moon formed by such an impact, it is likely that other inner planets also would have been subjected to comparable impacts.
If G arises as the group of units of a ring A, then an inner automorphism on G can be extended to a projectivity on the projective space over A by inversive ring geometry.
If the rainwater overflows the graduated inner cylinder, the larger outer container will catch it.
If W = V, then one can also pair the covector w * ∈ V * with the vector v ∈ V via ( w *, v ) → w *( v ), which is the duality pairing between V and its dual, sometimes called the inner product.
If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold.
If the cage is grounded, the excess charges will go to the ground instead of the outer face, so the inner face and the inner charge will cancel each other out and the rest of the cage will retain a neutral charge.
If the spacecraft's purpose is only to fly by the inner planet, then there is typically no need to slow the spacecraft.
If they fall from the utricle and become loose in the semicircular canals, they can distort the sense of movement and cause a mismatch between actual head movement and the information sent to the brain by the inner ear, causing a spinning sensation.
If any humidity subsists in the compressed air, the engine will stop due to inner icing.
If the star is in a binary system then the material will fall in through the inner Lagrangian point.
If and automorphism
If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.
If, then this is a relation-preserving automorphism.
A representation of a Lie group G on a vector space V ( over a field K ) is a smooth ( i. e. respecting the differential structure ) group homomorphism G → Aut ( V ) from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into GL ( n, K ).
If A is a subalgebra of B, then for every invertible b in B the function which takes every a in A to b < sup >− 1 </ sup > a b is an algebra homomorphism ( in case, this is called an inner automorphism of B ).
If the automorphism involved is the identity, then the reciprocity is called a projective correlation.
If are two duality transforms then their composition is an order automorphism of S ; thus, any two duality transforms differ only by an order automorphism.
If the target Y is commutative, then an antihomomorphism is the same thing as a homomorphism and an antiautomorphism is the same thing as an automorphism.
If M is an oriented manifold, Aut ( M ) would be the orientation-preserving automorphisms of M and so the mapping class group of M ( as an oriented manifold ) would be index two in the mapping class group of M ( as an unoriented manifold ) provided M admits an orientation-reversing automorphism.
If the first equation is relaxed to just have, then f is a partial automorphism ( with inverse g ).
If and group
`` If the day should ever come that foreign invaders swarm ashore along the Gulf Coast '', the account reads, `` they can count on heavy opposition from a group of commando-trained telephone employees -- all girls.
If coffee is sold at the cafeteria, let a few workers in each department get it for the whole group.
If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.
If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.
If the joining of monomers takes place at such a carbon atom, the free hydroxy group of the pyranose or furanose form is exchanged with an OH-side-chain of another sugar, yielding a full acetal.
If the amino group is removed from an amino acid, it leaves behind a carbon skeleton called an α-keto acid.
If a bubble touches identically-colored bubbles, forming a group of three or more, those bubbles — as well as any bubbles hanging from them — are removed from the field of play, and points are awarded.
If the Bastarnae remained an identifiable group, it is highly likely that they participated in the vast Gothic-led migration, driven by Hunnic pressure, that was admitted into Moesia by emperor Valens in 376 and eventually defeated and killed Valens at Adrianople in 378.
* 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 ).
If the carbonyl group is an aldehyde, the monosaccharide is an aldose ; if the carbonyl group is a ketone, the monosaccharide is a ketose.
If there is a tie, we must consider the atoms at distance 2 from the stereocenter — as a list is made for each group of the atoms bonded to the one directly attached to the stereocenter.
If the addition of a functional group takes place at the less substituted carbon atom of the double bond, then the electrophilic substitution with acids is not possible.
If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.
If G is a group, and g is a fixed element of G, then the conjugation map
If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in " standard form " meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1.
If a database cannot tolerate even such smaller probability of data loss, then the RAID group itself is replicated ( mirrored ).
If a database cannot tolerate even such smaller probability of data loss, then the RAID group itself is replicated ( mirrored ).
* If we think of as the group of real numbers under addition, then the direct product still consists of.
If the ether is part of a more complex molecule, it is described as an alkoxy substituent, so-OCH < sub > 3 </ sub > would be considered a " methoxy -" group.
If there is another functional group at a carbon, it may be named with the Greek letter, e. g., the gamma-amine in gamma-aminobutanoic acid is on the third carbon of the carbon chain attached to the carboxylic acid group.
Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.
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