Metropolitan France covers an area of 551,695 km² (213,011 sq. miles). Excluding the territory of Adélie Land in Antarctica, this is 81.8% of the nation's territory. As of January 1, 2011, 63,136,180 people lived in metropolitan France. This is 95.9% of the nation's population.
The five overseas departments – Martinique, Guadeloupe, Réunion, French Guiana, and Mayotte – have the same political status as the departments in metropolitan France. Metropolitan France and these five overseas departments together are sometimes called "France entière" ("entire France") by the French administration, especially by INSEE. This grouping does not include the French overseas collectivities and territories, which have more autonomy than the overseas departments.
Overseas France
Overseas France () is the part of France that is outside of the European continent. It consists of all overseas departments, territories and collectivities. These territories have several different legal statuses and levels of autonomy. Overseas France includes island territories in the Atlantic, Pacific and Indian oceans, French Guiana in South America, and Adélie Land in Antarctica. Each inhabited territory is represented in both the French National Assembly and the French Senate (which together make up the Parliament of France).
2,685,705 people lived in the overseas departments and territories in January 2011.
Overseas departments and regions are completely part of France.
The category of "overseas collectivity" was created by changes made to France's constitution in 28 March 2003. Each overseas collectivity has its own written laws.
The overseas departments and territories are represented by 27 "députés" in the French National Assembly and 21 senators in the French Senate. This is 4.7% of the 577 "députés" and 6% of the 343 senators.
The 11 French Overseas Territories are :
Lands generally uninhabited, except by researchers in scientific stations.
Justin Raimondo
Justin Raimondo (born Dennis Raimondo; November 18, 1951 – June 27, 2019) was an American writer and the editorial director of the website Antiwar.com. He was born in White Plains, New York.
Raimondo died on June 27, 2019 in Sebastopol, California from lung cancer at the age of 67.
Group (mathematics)
In mathematics, a group is a kind of algebraic structure. A group is a set with an operation. The group's operation shows how to replace any two elements of the group's set with a third element from the set in a useful way. A familiar example of a group is the set of integers with the addition operation.
Instead of "an element of the group's set", mathematicians usually save words by saying "an element of the group".
Mathematicians use capital letters to stand for groups. They often use "G", "H", or "K".
They also use lower-case letters to stand for group elements. For example, they would say ""a" is in "G"" to mean ""a" is an element of "G"".
They write group operations with symbols like • or *, or by writing two elements next to each other. So ""a" • "b"", ""a" * "b"", and ""ab"" can all mean "the element formed when the group's operation combines "a" and "b"".
Not every set and operation make a group. A group's set and operation must obey some special rules. These are called group axioms. The following list has each axiom in both words and mathematical symbols.
To name the group, one should say all three parts: ("G", •, "e"). But when it is clear, one may just say that "G" is a group.
It is possible to find that for two elements "a" and "b" in a group, "a" • "b" is different from "b" • "a". For example, maybe "a" • "b" = c, but "b" • "a" = d.
For some groups, however, "a" • "b" = "b" • "a" for every "a" and "b" in the group. These special groups have a special name. They are called Abelian groups and their operation is said to be commutative"."
The number of elements in a group is called the group's "order".
One everyday example of a group is the set of whole numbers (integers) with the addition operation. It is easy to see that when adding two integers, the result is always an integer. So closure is true. The identity element for this group is zero. Because the order of additions does not matter (in other words, formula_1), associativity rule is also true. And the inverse of any integer is its opposite number (the inverse element of zero is zero). This group is also an abelian group. It is an infinite group (also called a group of "infinite order") because it has an infinite number of elements: all the integers.
The set of integers with the multiplication operation is not a group, because with multiplication, most integers have no inverse that is also an integer (closure rule is not always obeyed).
Another important kind of groups are groups of symmetries. Symmetry groups provide many examples of groups. It can be proved that every group of finite order is a group of symmetries. For this reason, group theory is called the mathematical study of symmetry.
Objects are symmetric if keep some interesting features when changed. Those changes of objects (of any kind) are called symmetries.
Sets have only elements in them so they are one of the simplest objects to think of. Symmetries of a set are one-to-one functions that shuffle the elements of this set. Those functions change elements one with other but for example keep their amount the same as before shuffling.
When such group is finite (has finite order), say it has "n" elements, one can think of this group as the set of those shuffles (one-to-one functions) of "n" elements. The symmetric group given on "n" elements is often written as formula_2.
Shuffles can be combined one after another—function composition is the "operation" of this group.
Because making shuffling after shuffling is yet another shuffling, that means that there is "closure".
Also three shuffles can be combined in any of two ways: first two in the beginning to give a shuffle to combine with a third one, or last two in the beginning to give a shuffle to be combined with the first one ("associavity").
The "identity element" is the shuffle in which nothing changes.
Any shuffle can be undone by reversing the change. That reverse is the "inverse element".
Because order of shuffling matters—it is important which of the shuffles go first—group of symmetries are almost never abelian.
Think of a square and (one-to-one) functions can that shuffle the place of points of the square but keep the square as a whole in one place. Interesting features are size and place of the square. Such symmetries are called (global) isometries (see congruence).
Here are all of the isometries of a square: turn the square clockwise 90, 180, or 270; turn the square counter-clockwise 90, 180, or 270; reflect the image of the square in a vertical mirror, or in a horizontal mirror, or a diagonal mirror to the right or the left; or leave the square as it was.
This is a "finite" group, or a group of (finite) order 8, because there are 8 different isometries in total. It is a non-abelian group because it makes a difference which order when making a turn and a reflection.
This is not only kind of symmetry for this square. When position is not an interesting feature then there are many more symmetries. Then infinitely many moves are allowed. If the shape of a square is all that matters then there is even more symmetries of this square, those are similarities.
This shows that it is important to say what one exactly mean by ‘symmetry’. What are interesting features of objects that keep the same during a change.
Idea to study geometry by studying symmetries of figures goes back to XIX century. It is how mathematicians think of geometry today. They even use those methods when studying different objects and speak about ‘geometry’ and ‘symmetries’ of strange objects.