If S and T are subgroups of G their product need not be a subgroup.

If G is a finite group and S and T are subgroups of G, then ST is a subset of G of size | ST | given by the product formula:

In particular, if S and T ( subgroups now ) intersect only in the identity, then every element of ST has a unique expression as a product st with s in S and t in T. If S and T also permute, then ST is a group, and is called a Zappa-Szep product.

* If H and K are subgroups of G, then

If c < sub > 1 </ sub > ∈ C is another pre-image of x in C then the subgroups p < sub >#</ sub > ( π < sub > 1 </ sub >( C, c )) and p < sub >#</ sub > ( π < sub > 1 </ sub >( C, c < sub > 1 </ sub >)) are conjugate in π < sub > 1 </ sub >( X, x ) by p-image of a curve in C connecting c to c < sub > 1 </ sub >.

If a composition series exists for a group G, then any subnormal series of G can be refined to a composition series, informally, by inserting subgroups into the series up to maximality.

If we deal only with London, then there will be smaller subgroups such as the Turkish community or teenagers ( just to give two examples ) who will mix with each other more than people outside their group.

If C < sub > n </ sub > denotes the cyclic group of order n, then the possible torsion subgroups are C < sub > n </ sub > with 1 ≤ n ≤ 10, and also C < sub > 12 </ sub >; and the direct sum of C < sub > 2 </ sub > with C < sub > 2 </ sub >, C < sub > 4 </ sub >, C < sub > 6 </ sub > or C < sub > 8 </ sub >.

If a simple group is not of Lie type over a field of given characteristic p, then the p-local subgroups usually have components in the generalized Fitting subgroup, though there are many exceptions for groups that have small rank, are defined over small fields, or are sporadic.

If the subgroups are more generally indexed by ordinal numbers, one obtains a transfinite series, such as this ascending series:

If all maximal subgroups are type I then an argument similar to the CN case shows that the group G cannot be an odd-order minimal simple group, so there are exactly two classes of maximal subgroups of types II, III, IV or V. Most of the rest of the proof now focuses on these two types of maximal subgroup S and T and the relation between them.

* If a group of odd order has no elementary abelian subgroups of rank 3, then its derived group is nilpotent.

If the study did not need and / or use a randomization procedure, one should check the success of the non-random sampling, for instance by checking whether all subgroups of the population of interest are represented in sample.

If the sample sizes in the two groups being compared are equal, Student's original t-test is highly robust to the presence of unequal variances.

If there are k samples with size and sample variances then Bartlett's test statistic is

If word classes differ in their resistance or liability to stem replacement within meaning slot, it is conceivable that individual meanings also differ with fair consistence trans-lingually.

If two groups differ only in isotopes, atomic masses are used at each step to break ties in atomic number.

If solutions that differ only by symmetry operations ( rotations and reflections ) of the board are counted as one, the puzzle has 12 unique ( or fundamental ) solutions.

If a commodity's value is not the same as its price, and therefore the magnitudes of each likely differ, then what is the relation between the two, if any?

If two numbers differ by one order of magnitude, one is about ten times larger than the other.

If they differ by two orders of magnitude, they differ by a factor of about 100.

If they differ by the wave is circularly polarized.

If outcomes of different runs of an experiment differ in any way that matters, they are distinct outcomes.

If all atoms were the same size, molar and volumetric heat capacity would be proportional and differ by only a single constant reflecting ratios of the atomic-molar-volume of materials ( their atomic density ).

If they do, why do they differ?

If the law of one price is not true for a certain commodity, the price levels will not differ enough from the level predicted by PPP.

If grammars differ only by having different rankings of CON, then the set of possible human languages is determined by the constraints that exist.

: If two functions have the same derivative on an interval, they must differ by a constant,

If the image data is not properly processed during sampling or reconstruction, the reconstructed image will differ from the original image, and an alias is seen.

If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects ' rest masses.

If the body is given an initial velocity at A, or if friction is taken into account, the curve that minimizes time will differ from the one described above.

If they do not differ, then they too are relative ; but if they differ, then, since everything which differs is relative to something ..., things which exist absolutely are relative.

If the two ratings differ by more than one point, another evaluation by an expert reader is required to resolve the discrepancy and determine the final score.

* If the amounts of the itemized deductions and the standard deduction do not differ much, the taxpayer may take the standard deduction to reduce the possibility of adjustment by the Internal Revenue Service ( IRS ).

If the majority of elements differ from zero, then it is common to refer to the matrix as a dense matrix.

" If they are separable, function will differ from original reality and exist independently, and in that way function will have its own original reality.

If the target can change its position significantly during the 30 minutes the missile is in the air on its way, the probability of the missile destroying the target is drastically reduced.

If, on the other hand, the project is only 10 % complete at week 8, the project is significantly over budget and behind schedule.

If results were left unadjusted for factors such as race, gender, and free or reduced price lunch program eligibility, private schools performed significantly better than public schools.

If the estimate of were not significantly different from 0, we would fail to find evidence that changes in the growth rate and unemployment rate were related.

If a more sophisticated data structure ( e. g., heap or binary tree ) is used, the time required for searching and insertion can be reduced significantly ; this is the essence of heap sort and binary tree sort.

If the fluid being measured is significantly dense, hydrostatic corrections may have to be made for the height between the moving surface of the manometer working fluid and the location where the pressure measurement is desired except when measuring differential pressure of a fluid ( for example across an orifice plate or venturi ), in which case the density ρ should be corrected by subtracting the density of the fluid being measured.

If each satellite had two dozen lasers, two dozen satellites on-station would significantly blunt any attack.

If the path length in the birefringent medium is sufficient, plane waves will exit the material with a significantly different propagation direction, due to refraction.

If events should deviate significantly, the story may then fall under the genre of alternate history, which is known for speculating on what could have happened if a significant historical event had occurred differently.

If the cooperative has significantly less of its product to sell due to weather or insects, it makes up for that loss with a profit on the markets, since the overall supply of the crop is short everywhere that suffered the same conditions.

If a surface heating the liquid is significantly hotter than the liquid then film boiling will occur, where a thin layer of vapor, which has low thermal conductivity, insulates the surface.

If choking has begun, the host plant must be pruned significantly lower than the dodder, as dodder is versatile and can grow back if present from haustoria.

If this would be the case, both the cost and complexity of any protective systems needed to allow the perpetrator to survive long enough to both build the bomb and carry out the attack, would be significantly reduced.

If, then is significantly larger than the sampling interval, and.

If, then is significantly smaller than the sampling interval, and.

If the economy were to improve significantly, there was need for a large injection of capital — a resource that could only be obtained from international financial institutions of the West.

If one or more links in the chain are missing or delayed, then the chances of survival drop significantly.

If treated early, there is a significantly higher chance of successful recovery.

If a commercial company were able to be more efficient, costs could come down significantly.

If the corset was worn loosely, it was possible to leave the lacing as adjusted and take the corset on and off using the front opening ( if the corset is worn snugly, this method will damage the busk if the lacing is not significantly loosened beforehand ).

If the patient were completely non-responsive to the maximal recommended dose of sertraline ( 200 mg ), further increasing the dose did not significantly improve the response rates.

* Cost and demand differences between firms: If costs vary significantly between firms, it may be impossible to establish a price at which to fix output.

If the interest were only to compound yearly at 6 %, the monthly payment would be significantly different.

If output impedance is large compared to the impedance of the headphones, significantly higher distortion will be present.