QUES: Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

What describes finite simple groups as the building pieces for all finite groups?
What is the answer?
ANS: The Jordan–Hölder theorem
QUES: The climate in Southeast Alaska is a mid-latitude oceanic climate (Köppen climate classification: Cfb) in the southern sections and a subarctic oceanic climate (Köppen Cfc) in the northern parts. On an annual basis, Southeast is both the wettest and warmest part of Alaska with milder temperatures in the winter and high precipitation throughout the year. Juneau averages over 50 in (130 cm) of precipitation a year, and Ketchikan averages over 150 in (380 cm). This is also the only region in Alaska in which the average daytime high temperature is above freezing during the winter months.

How much precipitation does Ketchikan receive each year?
What is the answer?
ANS: over 150 in (380 cm)
QUES: As Western Europe witnessed the formation of new kingdoms, the Eastern Roman Empire remained intact and experienced an economic revival that lasted into the early 7th century. There were fewer invasions of the eastern section of the empire; most occurred in the Balkans. Peace with Persia, the traditional enemy of Rome, lasted throughout most of the 5th century. The Eastern Empire was marked by closer relations between the political state and Christian Church, with doctrinal matters assuming an importance in eastern politics that they did not have in Western Europe. Legal developments included the codification of Roman law; the first effort—the Theodosian Code—was completed in 438. Under Emperor Justinian (r. 527–565), another compilation took place—the Corpus Juris Civilis. Justinian also oversaw the construction of the Hagia Sophia in Constantinople and the reconquest of North Africa from the Vandals and Italy from the Ostrogoths, under Belisarius (d. 565). The conquest of Italy was not complete, as a deadly outbreak of plague in 542 led to the rest of Justinian's reign concentrating on defensive measures rather than further conquests. At the emperor's death, the Byzantines had control of most of Italy, North Africa, and a small foothold in southern Spain. Justinian's reconquests have been criticised by historians for overextending his realm and setting the stage for the Muslim conquests, but many of the difficulties faced by Justinian's successors were due not just to over-taxation to pay for his wars but to the essentially civilian nature of the empire, which made raising troops difficult.

In what century was there peace with one of Rome's traditional enemies?
What is the answer?
ANS:
5th