point group

Every molecule has a complete, finite list of moves that leave it looking unchanged — its symmetry operations. Collect that whole list into one bundle and give it a short name, and you have the molecule's point group. It is like a passport stamp: tell another chemist a molecule is "C2v" or "Td" and you have handed over its entire symmetry profile in a couple of characters.

The set is called a group because it obeys the rules mathematicians require: doing any two operations in a row gives a result that is itself in the set (closure), the identity E does nothing, every operation has an inverse that undoes it, and the order of combining is associative. It is called a point group (rather than a space group, used for crystals) because at least one point — usually the center of the molecule — stays fixed under every operation. The common molecular point groups come with standard Schoenflies labels: C2v (water), C3v (ammonia), D3h (BF3, flat trigonal), Td (methane, tetrahedral), Oh (SF6, octahedral), D-infinity-h (linear symmetric like CO2), and C-infinity-v (linear unsymmetric like CO), among others.

Naming the point group is not bookkeeping for its own sake — it is the key that unlocks the character table for that group, and from there everything follows by rule rather than guesswork. Once you know a complex is Oh, you can immediately say how its d orbitals split (into t2g and eg sets), which of its vibrations will appear in the infrared or Raman spectrum, whether it can have a dipole moment, and which ligand orbital combinations are allowed to bond with which metal orbitals. The point group is the organizing principle of structural inorganic chemistry.