* Let H be a group, and let G be the direct product H × H. Then the subgroups

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Then Exxon CEO, Lawrence G. Rawl, shaped the company's response.

Let a, b, and c be elements of G. Then:

Then and become partially defined operations on G, and will in fact be defined everywhere ; so we define * to be and to be.

Then with the dawn of new technologies such as electricity, the telegraph, and new forms of powered transportation, writers including Jules Verne and H. G. Wells created a body of work that became popular across broad cross-sections of society.

Then each element in G is represented in some way by a product

⟨ H ⟩, be the group generated by H. Then the word problem in H < sup >*</ sup > is solvable: given two words h, k in the generators H of H < sup >*</ sup >, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution the word problem for the class K ( say ) of finitely generated groups that can be embedded in G. If this were the case the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers.

Then, in February 1905, his eldest son W. G.

Let S be a subgroup of G, and let N be a normal subgroup of G. Then:

Then, given any point x and neighbourhood G of x, there is a closed neighbourhood E of x that is a subset of G.

Then F and G can be restricted to D < sub > 1 </ sub > and C < sub > 1 </ sub > and yield inverse equivalences of these subcategories.

Then R (( G )) is the ring of formal power series on G ; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same.

Then you type the first component, which is also ' Y ' for the 点 stroke, then a ' G ' for the 横 stroke ， and since you now already have three strokes, you type the last stroke, which also happens to be a 捺, arriving at the keycode ' YYGY ' for the complete character.

For such a subgroup G, define Fix ( G ) to be the field consisting of all elements of L that are held fixed by all elements of G. Then the maps E Gal ( L / E ) and G Fix ( G ) form an antitone Galois connection.

Then he would get to his feet, as though rising in honor of his own remarkable powers, and say almost invariably, `` Gentlemen, this is an amazing story!!

Then, Jesus indicated that God's forgiveness is unlimited.

Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.

Then it added: `` It is not possible to determine how extensive these ill effects will be -- nor how many people will be affected ''.

Then the words fell into a pattern: `` Mollie the Mutton is scratching her nose, Scratching her nose in the rain.

Then he thought of Aaron Blaustein standing in his rich house saying: `` God is tired of taking the blame.

Then it is marked on the inside where it comes in contact with the transom, frames, keelson and all the battens.

Then it is replaced and fastened.

Then the chines are rounded off and the bottom is rough-sanded in preparation.

Then, a group of eggs is deposited in a cavity in the beebread loaf and the egg compartment is closed.

Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Then, too, the utmost clinical flexibility is necessary in judiciously combining carefully timed family-oriented home visits, single and group office interviews, and appropriate telephone follow-up calls, if the worker is to be genuinely accessible and if the predicted unhealthy outcome is to be actually averted in accordance with the principles of preventive intervention.

Then the editorial added prophetically: `` how far they may reach in Asia is yet undetermined, but they fall far short of our dreams of the war conferences ''.

Then she catapults into `` everything and everybody '', putting particular violence on `` everybody '', indicating to the linguist that this is a spot to flag -- that is, it is not congruent to the patient's general style of speech up to this point.

Then comes the time when the last wire is removed and Susie walks out a healthier and more attractive girl than when she first went to the orthodontist.

Then, with the new affluence, there is actually a sallying forth into the wide, wide world beyond the precincts of New York.

Then, if the middle number is activated to its greatest potential in terms of this square, through multiplying it by the highest number, 9 ( which is the square of the base number ), the result is 45 ; ;

Then the figures which held his attention became a group of shattered trees, standing like the grotesques of a medieval damnation scene.

Then suddenly a group of men and dogs circled him.

Then, in order to prevent Polk from dissipating his forces by implementing his proposal to allow some men to join a partisan group, Johnston ordered him to send Pillow and 5, 000 men to Fort Donelson.

Then the required information is collected from a simple random sample of the elements within each selected group.

Then the lists are compared atom by atom ; at the earliest difference, the group containing the atom of higher atomic number receives higher priority.

Then, the leaving group, usually a proton, is split off and the aromaticity is restored.

Then it took 3 to 6 millions of years for the crown group to differentiate in Mid Cretaceous.

Then is a group whose identity element is The group inverse of an arbitrary group element is the function inverse

Then in the early 1980s there were major changes in the Australian brewing industry, including the merger of Castlemaine ( Brisbane ), Swan ( Perth ) and Toohey's ( Sydney ) into a national brewing group, as a result of acquisitions by Perth entrepreneur Alan Bond.

The character and themes from the comic strip Little Nemo were used in a song " Scenes from a Night's Dream " written by Phil Collins and Tony Banks of the progressive rock group Genesis on their 1978 recording, ... And Then There Were Three ... Another progressive rock group, from Germany, called Scara Brae also recorded a musical impression of the comic on their rare self titled disc from 1981 ( the track was actually recorded 2 years earlier ).

Then in 1965, a group in Atlanta applied for franchises in both the American Football League and the NFL, acting entirely on its own with no guarantee of stadium rights.

Then there are operators to rename columns, and summarizing or aggregating operators, and if you permit relation values as attributes ( RVA – relation-valued attribute ), then operators such as group and ungroup.

Then the equivalence class of the identity is the symmetry group of the figure, and every equivalence class corresponds to one isomorphic version of the figure.

In 1986, a screening of the entire Monkees television series by MTV led to renewed interest in the group, followed by a single (" That Was Then, This Is Now " reached number 20 on the Billboard Hot 100 in the U. S .), a 20th Anniversary Tour, a greatest hits album and a brand new LP, Pool It!

Then the subjects meet with a group of experts, who give the subjects instructions on how to improve things ; they offer aid and encouragement along the way.

Then in 630, Bishop Cyrus was made Patriarch of Alexandria, and he soon won over another Non-Chalcedonian group.