For the family is the simplest example of just such a unit, composed of people, which gives us both some immunity from, and a way of dealing with, other people.

For answers to such questions we must turn to the anthropologists, the biologists, the historians, the psychologists, and the sociologists.

For my part I find it difficult to conceive such a state of affairs.

For a while there was such shrill girlish commotion I couldn't have made myself heard if I'd had the equivalent of the message to Garcia.

For the most part, however, the new version is contemporary and, as such, should be the means for many to attain a clearer comprehension of the meaning of those words recorded so many hundreds of years ago by the first followers of Christ.

For it is such a distinguished place, with such fine works of art and such a big library, that there can be little doubt but that the owner has become depraved by all this culture.

For the oyabun to make such a trip was either a sign of great weakness or an indication of equally great confidence, and from all the available information it was probably the latter.

For what Sam Rayburn's life in this House teaches us is that loyalty and character are not divisive and there is no such thing as being for your country and neglecting your district.

For United States expenditures under subsections ( A ), ( B ), ( D ), ( E ), ( F ), ( H ) through ( R ) of Section 104 of the Act or under any of such subsections, the rupee equivalent of $200 million.

For grant to the Government of India under subsection ( E ) of Section 104 of the Act, the rupee equivalent of not more than $538 million for financing such projects to promote balanced economic development as may from time to time be mutually agreed.

For loan to the Government of India under subsection ( G ) of Section 104 of the Act, the rupee equivalent of not more than $538 million for financing such projects to promote balanced economic development as may be mutually agreed.

You may apply for such an extension by filing Form 2688, Application For Extension Of Time To File, with the District Director of Internal Revenue for your district, or you may make your application in a letter.

For the making of selections on the basis of excellence requires that any foundation making the selections shall have available the judgments of a corps of advisors whose judgments are known to be good: such judgments can be known to be good only by the records of those selected, by records made subsequent to their selection over considerable periods of time.

For example, the interest of past members of the Foundation's Advisory Board remains such that they place their knowledge and judgments at our disposal much as they had done when they were, formally, members of that Board.

For instance, a site adjoining other publicly owned lands, such as a national forest or a public road, may be desirable, whereas a site next to an industrial plant might not.

For prevention of these diseases during periods of stress such as shipping, excessive handling, vaccination, extreme weather conditions: 350 milligrams per head per day for 30 days only.

For specialized types of displays, such as large three-dimensional units reproducing a product, package, human or animal figures, etc., reinforced plastics and rotationally molded vinyl plastisols are other materials frequently used.

For example, the marked susceptibility of the monkey to respiratory infection might be related to its delicate, long alveolar ducts and short, large bronchioles situated within a parenchyma entirely lacking in protective supportive tissue barriers such as those found in types 1, and 3.

For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square.

For if such were the case, either the plane of the two lines would meet **zg in more than K points or, alternatively, the order of the image regulus of the pencil determined by the two lines would be too high.

For the industry of this model, the effect of such public pressures in the past has been to hold the price well below the short-run profit-maximizing price ( given the wage rate and the level of GNP ), and even below the entry-limited price ( but not below average cost ).

For such an industry, it is only `` safe '' to raise its price if such an increase is manifestly `` justified '' by rising costs ( due to rising wages, etc. ).

for some natural number n. Moreover, since, the commutator subgroup is normal in G. For any homomorphism f: G → H,

For x in X, the vertex group consists of those ( g, x ) with gx = x, which is just the isotropy subgroup at x for the given action ( which is why vertex groups are also called isotropy groups ).

For any subgroup, the following conditions are equivalent to normality.

For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.

For instance, the normalizer N of a proper subgroup H of a finite p-group G properly contains H, because for any counterexample with H = N, the center Z is contained in N, and so also in H, but then there is a smaller example H / Z whose normalizer in G / Z is N / Z = H / Z, creating an infinite descent.

For general groups, Cauchy's theorem guarantees the existence of an element, and hence of a cyclic subgroup, of order any prime dividing the group order ; Sylow's theorem extends this to the existence of a subgroup of order equal to the maximal power of any prime dividing the group order.

For solvable groups, Hall's theorems assert the existence of a subgroup of order equal to any unitary divisor of the group order ( that is, a divisor coprime to its cofactor ).

( For the example of x + y − z, the subgroup H in S < sub > 3 </ sub > contains the identity and the transposition ( xy ).

For a nonabelian example, consider the subgroup of rotations of R < sup > 3 </ sup > generated by two rotations by irrational multiples of 2π about different axes.

For example, in any topological group the identity component ( i. e. the connected component containing the identity element ) is a closed normal subgroup.

for all g < sub > 1 </ sub >, g < sub > 2 </ sub >, h < sub > 1 </ sub >, h < sub > 2 </ sub > ∈ G. For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the cosets of this subgroup.

For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f: H → G is always a monomorphism ; but f has a left inverse in the category if and only if H has a normal complement in G.

For example, the point group of a molecule is a subgroup of O ( 3 ).

For any abelian group and any prime number p the set A < sub > Tp </ sub > of elements of A that have order a power of p is a subgroup called the p-power torsion subgroup or, more loosely, the p-torsion subgroup:

For each prime number p, this provides a functor from the category of abelian groups to the category of p-power torsion groups that sends every group to its p-power torsion subgroup, and restricts every homomorphism to the p-torsion subgroups.

For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G ; and G is said to be nilpotent of class n. ( By definition, the length is n if there are n + 1 different subgroups in the series, including the trivial subgroup and the whole group.

For each integer n > 1, the dicyclic group Dic < sub > n </ sub > can be defined as the subgroup of the unit quaternions generated by

For n > 1, there are additional conditions, i. e. Sp ( 2n, F ) is then a proper subgroup of SL ( 2n, F ).

In mathematics, specifically group theory, the index of a subgroup H in a group G is the " relative size " of H in G: equivalently, the number of " copies " ( cosets ) of H that fill up G. For example, if H has index 2 in G, then intuitively " half " of the elements of G lie in H. The index of H in G is usually denoted | G: H | or < nowiki ></ nowiki >.