Context and question: for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the etale fundamental group.
What group is an advanced observation of infinite field extensions and groups that is adapted for the needs of algebraic geometry?
Answer: the etale fundamental group
Context and question: In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right cosets of H containing g are
What should not be considered when asking if a reflection has been performed?
Answer: rotation operations
Context and question: Palermo is surrounded by mountains, formed of calcar, which form a cirque around the city. Some districts of the city are divided by the mountains themselves. Historically, it was relatively difficult to reach the inner part of Sicily from the city because of the mounts. The tallest peak of the range is La Pizzuta, about 1,333 m (4,373 ft.) high. However, historically, the most important mount is Monte Pellegrino, which is geographically separated from the rest of the range by a plain. The mount lies right in front of the Tyrrhenian Sea. Monte Pellegrino's cliff was described in the 19th century by Johann Wolfgang von Goethe, as "The most beautiful promontory in the world", in his essay "Italian Journey".
Where is Monte Pellegrino in relation to other peaks?
Answer:
geographically separated from the rest of the range by a plain