The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism is isomorphic to where ker ( φ ) denotes the kernel of φ.

Specifically, consider the group ( R < sup >+</ sup >,< big >×</ big >) of positive real numbers under ordinary multiplication.

Specifically, it is an Islamist separatist group based in and around the southern islands of the Republic of the Philippines, primarily Jolo, Basilan, and Mindanao.

Specifically, if L / K is a Galois extension, we consider the group G = Gal ( L / K ) consisting of all field automorphisms of L which keep all elements of K fixed.

Specifically, we perceive there to be a group of 36 circles on the left side of the image, and three groups of 12 circles on the right side of the image.

Specifically, a polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable.

Specifically, children in this group have greater levels of ADHD symptoms, neuropsychological deficits, more academic problems, increased family dysfunction, and higher likelihood of aggression and violence.

( Specifically, if the individual has a " good idea " that he might otherwise hold back from the group because it is too controversial, he will share that idea in this conversation.

Specifically, the condition provided by: a. a military or defense advantage over any foreign nation or group of nations ; b. a favorable foreign relations position ; or c. a defense posture capable of successfully resisting hostile or destructive action from within or without, overt or covert.

Specifically, let G be a simply-connected Lie group with Lie algebra.

Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group.

Specifically, if K / Q is a finite Galois extension then to each ( positive ) prime p which does not ramify in K and to each prime ideal P lying over p in K there is a unique element g of Gal ( K / Q ) satisfying the condition g ( x ) = x < sup > p </ sup > ( mod P ) for all integers x of K. Varying P over p changes g into a conjugate ( and every conjugate of g occurs in this way ), so the conjugacy class of g in the Galois group is canonically associated to p. This is called the Frobenius conjugacy class of p and any element of

Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and HNN extensions.

Specifically, for ( a projective line ), all points are collinear, so the collineation group is exactly the symmetric group of the points of the projective line, and except for and ( where PGL is the full symmetric group ), PGL is a proper subgroup of the full symmetric group on these points.

Specifically, let G be a group and let X and Y be two associated G-sets.

Specifically, it is an Islamist separatist group based in and around the southern islands of the Republic of the Philippines, primarily Jolo, Basilan, and Mindanao.

Specifically, the results of the study were that the mean weight loss of Weight Watchers participants was 2. 3 times more than ( 3. 3 times as much as ) the self-help group at one year, and essentially undefined at two years.

Specifically, inhabitants in the Frankston area were from the Bunurong language group, of the Mayone-bulluk clan.

Specifically, an amide results from an acid, in which a carbon atom is double bonded to oxygen and also to an hydroxyl group, when the hydroxyl group is replaced by an amine.

Specifically, valine CETP homozygotes but not heterozygotes experienced a relative 51 % less decline in memory compared to a reference group after adjusting for demographic factors and APOE status.

Specifically, the conceptual diagram graph 1 identifies only three boxes, two ellipses, and four arrows ( and their five labels ), whereas the picture 1 shows much more pictorial detail, with the scores of implied relationships as implicit in the picture rather than with the nine explicit details in the graph.

Specifically, they grow on chocolate agar with carbon dioxide.

Specifically it is the amount of a substance that contains as many elementary entities ( atoms, molecules or ions ) as there are atoms in 0. 012 kilogram ( or 12 grams ) of carbon-12, where the carbon-12 atoms are unbound, at rest and in their ground state.

Specifically, after acknowledging the various popular theories in vogue at the time, of how atoms were reasoned to attach to each other, i. e. " hooked atoms ", " glued together by rest ", or " stuck together by conspiring motions ", Newton states that he would rather infer from their cohesion, that " particles attract one another by some force, which in immediate contact is exceedingly strong, at small distances performs the chemical operations, and reaches not far from the particles with any sensible effect.

Specifically, they observed what they later named Non-Classical Rotational Inertia, an unusual decoupling of the solid helium from a container's walls which could not be explained by classical models but which was consistent with a superfluid-like decoupling of a small percentage of the atoms from the rest of the atoms in the container.

Specifically, Van der Waals strain is considered a form of strain where the interacting atoms are at least four bonds away from each other.

Specifically, Congress should consider authorizing the Peace Corps to receive contributions from American businesses, unions, civic organizations and the public at large.

Specifically, corporations are accused of seeking to maximize profit at the expense of work safety conditions and standards, labor hiring and compensation standards, environmental conservation principles, and the integrity of national legislative authority, independence and sovereignty.

Specifically, at speeds of OC-3 and above, the cost of segmentation and reassembly ( SAR ) hardware makes ATM less competitive for IP than Packet Over SONET ( POS ); because of its fixed 48-byte cell payload, ATM is not suitable as a data link layer directly underlying IP ( without the need for SAR at the data link level ) since the OSI layer on which IP operates must provide a maximum transmission unit ( MTU ) of at least 576 bytes.

Specifically it states that for any integers n ≥ 0 and m ≥ 1, the functions J < sub > n </ sub >( x ) and J < sub > n + m </ sub >( x ) have no common zeros other than the one at x = 0.

Specifically, they showed that for sufficiently large, there are at least Carmichael numbers between 1 and.

Specifically, by the Casorati – Weierstrass theorem, for any transcendental entire function f and any complex w there is a sequence with, is necessarily a polynomial, of degree at least n.

Specifically, the distribution of this trait suggests that this need may have ( initially ) grown so intense at certain ( early ) points that those among said ( initially more deeply pigmented skinned ) Northern-migrants with mutations for straighter hair survived and had children at ( somewhat ) higher rates.

Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v ^< sub > g </ sub > =

Specifically, umpires were now given the power — and the responsibility — to intervene if they considered a bowler was deliberately aiming at a batsman with intent to injure.

According to Stephen Suleyman Schwartz, in his book " The Two Faces of Islam ", “ some say that during this vagabondage Ibn Abdul Wahhab came into contact with certain Englishmen who encouraged him to personal ambition as well as to a critical attitude about Islam .” Specifically, Mir ’ at al Harramin, a Turkish work by Ayyub Sabri Pasha, written in 1888, states that in Basra, Abdul Wahhab had come into contact with a British spy by the name of Hempher, who “ inspired in him the tricks and lies that he had learned from the British Ministry of the Commonwealth .”

Specifically, Mica is unusual in that it is a good electrical insulator at the same time as being a good thermal conductor.

Specifically, McVeigh arranged the barrels in the shape of a backwards J ; he later said that for pure destructive power, he would have put the barrels on the side of the cargo bay closest to the Murrah Building ; however, such an unevenly distributed load might have broken an axle, flipped the truck over, or at least caused it to lean to one side, which could have drawn attention.

Specifically, the problem of combining quantum mechanics and gravity becomes an issue only at very high energies, and may well require a totally new kind of model.

Specifically, both rivers flow into Suisun Bay, which flows through the Carquinez Strait to meet with the Napa River at the entrance to San Pablo Bay, which connects at its south end to San Francisco Bay.

Specifically, the campaign raised $ 253. 7 million for undergraduate financial aid, as well as $ 2. 33 billion for its initiative in " Seeking Solutions " to global problems, $ 1. 61 billion for " Educating Leaders " by improving K-12 education, and $ 2. 11 billion for " Foundation of Excellence " aimed at providing academic support for Stanford students and faculty.

" Specifically, Magnotta and Strohl focus on The Marine Biologist, in which George becomes embroiled in yet another lie, and on The Red Dot in which George tries to save a few dollars at Elaine's expense by giving her a marked-down cashmere sweater.

Specifically, if v is a tangent vector of M at a point x ( thought of as a derivation ) then define the directional derivative in the direction v by