1. [n]  any number of entities (members) considered as a unit
2. [n]  a set that is closed, associative, has an identity element and every element has an inverse
3. [n]  (chemistry) two or more atoms bound together as a single unit and forming part of a molecule
4. [v]  arrange into a group or groups; "Can you group these shapes together?"
5. [v]  form a group or group together

1. \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
      plants, 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.
   
   

2. \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 more columns placed upon
      the same pedestal.
   
   

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.