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

Group \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.
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         The difficulty lies in drawing and disposing, or, as
         the painters term it, in grouping such a multitude of
         different objects.                       --Prior.
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   Grouped columns (Arch.), three or more columns placed upon
      the same pedestal.
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Group \Group\ (gr[=oo]p), n. [F groupe, It. gruppo, groppo,
   cluster, bunch, packet, group; of G. origin: cf. G. kropf
   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.
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   2. An assemblage of objects in a certain order or relation,
      or having some resemblance or common characteristic; as,
      groups of strata.
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   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.
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   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.
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Residue \Res"i*due\ (r?z"?-d?), n. [F. r['e]sidu, L. residuum,
   fr. residuus that is left behind, remaining, fr. residere to
   remain behind. See Reside, and cf. Residuum.]
   1. That which remains after a part is taken, separated,
      removed, or designated; remnant; remainder.
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            The residue of them will I deliver to the sword.
                                                  --Jer. xv. 9.
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            If church power had then prevailed over its victims,
            not a residue of English liberty would have been
            saved.                                --I. Taylor.
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   2. (Law) That part of a testeator's estate wwhich is not
      disposed of in his will by particular and special legacies
      and devises, and which remains after payment of debts and
      legacies.
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   3. (Chem.) That which remains of a molecule after the removal
      of a portion of its constituents; hence, an atom or group
      regarded as a portion of a molecule; a moiety or
      group; -- used as nearly equivalent to radical, but in
      a more general sense.
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   Note: The term radical is sometimes restricted to groups
         containing carbon, the term residue and moiety being
         applied to the others.
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   4. (Theory of Numbers) Any positive or negative number that
      differs from a given number by a multiple of a given
      modulus; thus, if 7 is the modulus, and 9 the given
      number, the numbers -5, 2, 16, 23, etc., are residues.
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   Syn: Rest; remainder; remnant; balance; residuum; remains;
        leavings; relics.
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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)