In chemistry, a group is a vertical column in the periodic table. Elements in the same group have similar properties, however, as one goes down a column, elements tend to increase in atomic radius and mass, and become more metallic.

Consider a nonempty set G with a binary operation @ such that for elements a and b in the group, the element a@b is also in G. G is a group if the following properties hold.

• The binary operation @ is associative. That is to say, for all elements a, b, c in G, it holds that a@(b@c) = (a@b)@c.
• There exists an identity element e such that a@e = e@a = a for all elements a in G.
• Every element a of G has an inverse, a-1 in G such that a@a-1 = a-1@a = e.

While I was in college, a fellow mathematics major and I made the conjecture that the set of all gross things formed a group (in TMA's sense, above). It seemed reasonable - the set was certainly closed under the operator of addition or mixing. For if you mix two gross things together, the result would have to be gross, right?

Alas, we were never able to prove our theory, and unfortunately we came to the conclusion that it is probably false. For we could not determine what the identity element of such a group would be. What could you add to anything gross that would not change it, yet was itself gross? The only candidate was water; but there are some things that cannot combine with anything to make water - for example, corned beef hash.

A side note for math geeks: although the set of gross things is not a group, the fact that it is closed under addition means that you could localize in the ring of all things, with addition being represented by formal sums and multiplication represented by mixing. Which would give you a fairly strange ring.

Group is also a sociological term. It is quite subjective and its exact definition changes with any change in the situation, but I shall try to illustrate its definition and usage to the best of my ability. A group is usually composed of two or more people, although in certain special cases it can be comprised of one person only, as we'll see shortly.

The scope of any given group is determined first and foremost by the members of said group and their willingness or lack thereof to include specific others within their group. An easy illustration of this phenomenon can be seen in a stage performance. Any time a person is onstage, the audience sees him or her as being divorced from them in the sense that he becomes part of a group that is mutually exclusive with the audience: the performers. If the performer leaves the stage and views the remainder of the show from the audience, acting like an audience member, he will most likely retain enough characteristics of a performer (i.e. costume, makeup, a memory in the audience's mind of his presence onstage) to preclude his acceptance as a full member of the audience. The audience finds it necessary to create its identity both through self-definition ("Audience members sit in chairs quietly.") as well as other-based definition ("Audience members are not performers."). He will probably be on the recieving end of behaviors that the audience normally reserves for persons who are not members of itself, such as stares, congratulations and awed handshakes. In other words, the audience maintains its membership and border integrity by failing to allow persons that it does not recognize as members into the group.

Not every group's boundaries are this rigid, however. On a bus, membership in the group of passengers is as simple as paying a fare. Once the basic requirements for membership in any group are met, whether those requirements have to do with a person's race, social standing, or ability to pay a dollar and a half to get on the bus, the person is almost always accepted as a full member of the group and becomes subject to its rights and obligations, which it is necessary to meet to remain a member of the group.

Another very important fact concerning groups is that every person has numerous group memberships at any given time. All of these memberships are, however, not equally relevant at any given time. For a man involved in the 1991 Los Angeles riots following the Rodney King beating, racial membership will be almost certainly the most important and relevant membership to him, while his status as a Rotary club member will probably be forgotten rather quickly. This does not mean that he is no longer a member of the Rotary Club group, but it does mean that racial category overshadows the Rotarians at that time in both the mind of the subject as well as the mind of the sociologist studying him. In this sense, the man can be taken to be a representative of said group, and especially if he is the only one of its members present, he can sometimes be said to constitute something of a group in himself when seen from an outside point of view. Reginald Denny, for instance, was the only white man present at his beating and therefore could be said to be a member of the "white group," although he is the only member present at the time. This is an especially vivid example of group definition by exclusion, as the group of black men in the news footage that were hitting him with bricks derived their group membership partially from contrast with the nonmember, to whom behavior was shown that completely precluded his possible membership in their group.

I believe that this covers most of the basic characteristics of groups, group formation, and group membership. I intend to talk about intergroup conflict, intragroup conflict, as well as a few other possibilities regarding special cases of group membership and definition in other nodes that I have not yet made. If there's anything else that you feel should appear in this writeup (as the possibility of my having missed an important something is quite distinct) please /msg me and I'll see what I can do to correct the omission.

In the game of Go (Wei Qi/Badouk), the term group can be used in one of two ways:

1. Strictly, in the sense of the rules, to mean several stones of the same colour which are all connected to one another along orthogonal (horizontal, vertical) directions. The stones in such a group will always live or be captured together; in order to take one stone prisoner, you must surround the whole group.

2. Loosely, and in a strategic sense, to mean several stones of the same colour in the same general area of the board. Although not a group according to the rules, since some stones can be captured without capturing the whole group, it is useful to think of them as such, because it is often possible to connect them together if so desired.

Consider the following diagram:

.........
.000.....
....0.0..
...0.....
.........

By the looser (strategic) definition, all these stones would be considered part of the same group, whereas, according to the "legal" definition, only the three in a line in the top left would be considered as such, while the others are merely individual stones.

Group (?), n. [F groupe, It. gruppo, groppo, cluster, bunch, packet, group; of G. origin: cf. G. krepf craw, crop, tumor, bunch. See Crop, n.]

1.

A cluster, crowd, or throng; an assemblage, either of persons or things, collected without any regular form or arrangement; as, a group of men or of trees; a group of isles.

2.

An assemblage of objects in a certain order or relation, or having some resemblance or common characteristic; as, groups of strata.

3. Biol.

A variously limited assemblage of animals or planta, having some resemblance, or common characteristics in form or structure. The term has different uses, and may be made to include certain species of a genus, or a whole genus, or certain genera, or even several orders.

4. Mus.

A number of eighth, sixteenth, etc., notes joined at the stems; -- sometimes rather indefinitely applied to any ornament made up of a few short notes.

Group, v. t. [imp. & p. p. Grouped (?); p. pr. & vb. n. Grouping.] [Cf. F. grouper. See Group, n.]

To form a group of; to arrange or combine in a group or in groups, often with reference to mutual relation and the best effect; to form an assemblage of.

The difficulty lies in drawing and disposing, or, as the painters term it, in grouping such a multitude of different objects. Prior.

Grouped columns Arch., three or moro columns placed upon the same pedestal.

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