Problem: Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that g • g = 1G for some element g of G is equivalent to proving that a(g) ∗ a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first.
What similar element do isomorphic groups carry?
The answer is the following: isomorphic groups

Problem: Homer refers to the "Hellenes" (/ˈhɛliːnz/) as a relatively small tribe settled in Thessalic Phthia, with its warriors under the command of Achilleus. The Parian Chronicle says that Phthia was the homeland of the Hellenes and that this name was given to those previously called Greeks (Γραικοί). In Greek mythology, Hellen, the patriarch of Hellenes, was son of Pyrrha and Deucalion, who ruled around Phthia, the only survivors after the great deluge. It seems that the myth was invented when the Greek tribes started to separate from each other in certain areas of Greece and it indicates their common origin. Aristotle names ancient Hellas as an area in Epirus between Dodona and the Achelous river, the location of the great deluge of Deucalion, a land occupied by the Selloi and the "Greeks" who later came to be known as "Hellenes". Selloi were the priests of Dodonian Zeus and the word probably means "sacrificers" (compare Gothic saljan, "present, sacrifice"). There is currently no satisfactory etymology of the name Hellenes. Some scholars assert that the name Selloi changed to Sellanes and then to Hellanes-Hellenes. However this etymology connects the name Hellenes with the Dorians who occupied Epirus and the relation with the name Greeks given by the Romans becomes uncertain. The name Hellenes seems to be older and it was probably used by the Greeks with the establishment of the Great Amphictyonic League. This was an ancient association of Greek tribes with twelve founders which was organized to protect the great temples of Apollo in Delphi (Phocis) and of Demeter near Thermopylae (Locris). According to the legend it was founded after the Trojan War by the eponymous Amphictyon, brother of Hellen.
Is the name more ancient than other group names in the area ?
The answer is the following: The name Hellenes seems to be older and it was probably used by the Greeks with the establishment of the Great Amphictyonic League.

Problem: In October 1745, Ewald Georg von Kleist of Pomerania, Germany, found that charge could be stored by connecting a high-voltage electrostatic generator by a wire to a volume of water in a hand-held glass jar. Von Kleist's hand and the water acted as conductors, and the jar as a dielectric (although details of the mechanism were incorrectly identified at the time). Von Kleist found that touching the wire resulted in a powerful spark, much more painful than that obtained from an electrostatic machine. The following year, the Dutch physicist Pieter van Musschenbroek invented a similar capacitor, which was named the Leyden jar, after the University of Leiden where he worked. He also was impressed by the power of the shock he received, writing, "I would not take a second shock for the kingdom of France."
Who first discovered the basic properties of capacitors?
The answer is the following: Ewald Georg von Kleist

Problem: Unwilling to accept and license Columbia's system, in February 1949 RCA Victor, in cooperation of its parent, the Radio Corporation of America, released the first 45 rpm single, 7 inches in diameter with a large center hole. The 45 rpm player included a changing mechanism that allowed multiple disks to be stacked, much as a conventional changer handled 78s. The short playing time of a single 45 rpm side meant that long works, such as symphonies, had to be released on multiple 45s instead of a single LP, but RCA claimed that the new high-speed changer rendered side breaks so brief as to be inaudible or inconsequential. Early 45 rpm records were made from either vinyl or polystyrene. They had a playing time of eight minutes.
What did RCA release to complete against Columbia's LP?
The answer is the following:
The 45 rpm player