ETYM F groupe, Italian gruppo, groppo, cluster, bunch, packet, group; of German origin: cf. German kropf craw, crop, tumor, bunch. Related to Crop.
1. A set that is closed, associative, has an identity element and every element has an inverse; SYN. mathematical group.
2. Any number of entities (members) considered as a unit; SYN. grouping.
3. Two or more atoms bound together as a single unit and forming part of a molecule; SYN. radical.

1. To arrange into a group or groups.
2. To form a group or group together; SYN. aggroup.

In mathematics, a finite or infinite set of elements that can be combined by an operation; formally, a group must satisfy certain conditions. For example, the set of all integers (positive or negative whole numbers) forms a group with regard to addition because: (1) addition is associative, that is, the sum of two or more integers is the same regardless of the order in which the integers are added; (2) adding two integers gives another integer; (3) the set includes an identity element 0, which has no effect on any integer to which it is added (for example, 0 + 3 = 3); and (4) each integer has an inverse (for instance, 7 has the inverse -7), such that the sum of an integer and its inverse is 0. Group theory is the study of the properties of groups.

In chemistry, a vertical column of elements in the periodic table. Elements in a group have similar physical and chemical properties; for example, the group I elements (the alkali metals: lithium, sodium, potassium, rubidium, cesium, and francium) are all highly reactive metals that form univalent ions. There is a gradation of properties down any group: in group I, melting and boiling points decrease, and density and reactivity increase.

In an organization, a number of people who work together. In a formal group, each person has a specific role allocated within the organization (for example, one worker might be the managing director, another the finance director, and another the sales manager). An informal group is a group of people who join together outside the formal structures of the organization. It could be workers who play squash together, or people who share crčche facilities.
Equally, informal groups may cut across formal structures in production and management. For example, two or three people may effectively run a department even though none of them is actually the head of department. Informal groups can be very threatening to people who believe that an organization should be run by formal groups and through formal structures.

A collection of elements that can be treated as a whole, such as a collection of records in a database report, or a collection of objects that can be moved and transformed as a single object in a drawing program. In various multiuser operating systems, a group is a set of user accounts, sometimes called members; privileges can be specified for the group, and each member will then have those privileges. See also built-in groups. Compare local group, user account.

In a drawing program, to transform a number of objects into a group. See also drawing program.