group

adj : for or by a group rather than individuals; "dipping each his

bread into a communal dish of stew"- Paul Roche; "a

communal settlement in which all earnings and food were

shared"; "a group effort" [syn: {communal}, {group(a)}]

n 1: any number of entities (members) considered as a unit [syn:

{grouping}]

2: two or more atoms bound together as a single unit and

forming part of a molecule [syn: {radical}]

3: a set that is closed, associative, has an identity element

and every element has an inverse [syn: {mathematical group}]

v 1: arrange into a group or groups; "Can you group these shapes

together?"

2: form a group or group together [syn: {aggroup}]

group

A group G is a non-empty {set} upon which a {binary} operator

▫ is defined with the following properties for all a,b,c in G:

Closure: G is closed under ▫, a▫b in G

Associative: ▫ is associative on G, (a▫b)▫c ◦ a▫(b▫c)

Identity: There is an identity element e such that

a▫e ◦ e▫a ◦ a.

Inverse: Every element has a unique inverse a' such that

a ▫ a' ◦ a' ▫ a ◦ e. The inverse is usually

written with a superscript -1.

(1998-10-03)