Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
If it is possible to answer this question, answer it for me (else, reply "unanswerable"): What are groups consisting of symmetries of given arithmetic concepts?
Ah, so.. Symmetry groups

Faced with the stripping of their powers, Wu Sangui, later joined by Geng Zhongming and by Shang Kexi's son Shang Zhixin, felt they had no choice but to revolt. The ensuing Revolt of the Three Feudatories lasted for eight years. Wu attempted, ultimately in vain, to fire the embers of south China Ming loyalty by restoring Ming customs, ordering that the resented queues be cut, and declaring himself emperor of a new dynasty. At the peak of the rebels' fortunes, they extended their control as far north as the Yangtze River, nearly establishing a divided China. Wu then hesitated to go further north, not being able to coordinate strategy with his allies, and Kangxi was able to unify his forces for a counterattack led by a new generation of Manchu generals. By 1681, the Qing government had established control over a ravaged southern China which took several decades to recover. Manchu Generals and Bannermen were initially put to shame by the better performance of the Han Chinese Green Standard Army, who fought better than them against the rebels and this was noted by Kangxi, leading him to task Generals Sun Sike, Wang Jinbao, and Zhao Liangdong to lead Green Standard Soldiers to crush the rebels. The Qing thought that Han Chinese were superior at battling other Han people and so used the Green Standard Army as the dominant and majority army in crushing the rebels instead of Bannermen. Similarly, in northwestern China against Wang Fuchen, the Qing used Han Chinese Green Standard Army soldiers and Han Chinese Generals such as Zhang Liangdong, Wang Jinbao, and Zhang Yong as the primary military forces. This choice was due to the rocky terrain, which favoured infantry troops over cavalry, to the desire to keep Bannermen in the reserves, and, again, to the belief that Han troops were better at fighting other Han people. These Han generals achieved victory over the rebels. Also due to the mountainous terrain, Sichuan and southern Shaanxi were also retaken by the Han Chinese Green Standard Army under Wang Jinbao and Zhao Liangdong in 1680, with Manchus only participating in dealing with logistics and provisions. 400,000 Green Standard Army soldiers and 150,000 Bannermen served on the Qing side during the war. 213 Han Chinese Banner companies, and 527 companies of Mongol and Manchu Banners were mobilized by the Qing during the revolt. 400,000 Green Standard Army soldiers were used against the Three Feudatories besides 200,000 Bannermen.
If it is possible to answer this question, answer it for me (else, reply "unanswerable"): How long did the Revolt of the Three Feudatories last?
Ah, so.. eight years

Chinese character dictionaries often allow users to locate entries in several ways. Many Chinese, Japanese, and Korean dictionaries of Chinese characters list characters in radical order: characters are grouped together by radical, and radicals containing fewer strokes come before radicals containing more strokes (radical-and-stroke sorting). Under each radical, characters are listed by their total number of strokes. It is often also possible to search for characters by sound, using pinyin (in Chinese dictionaries), zhuyin (in Taiwanese dictionaries), kana (in Japanese dictionaries) or hangul (in Korean dictionaries). Most dictionaries also allow searches by total number of strokes, and individual dictionaries often allow other search methods as well.
If it is possible to answer this question, answer it for me (else, reply "unanswerable"): What is grouped together by radicals?
Ah, so..
characters