Group theory

In mathematics and abstract algebra, group theory studies algebraic structures known as groups. Several physical systems, like crystals and the hydrogen atom, can be modeled by symmetry groups.

Representation of groups: Saying that a group G acts on the set X means that every element defines a bijective map on a set in a way compatible with group structure. When X has more structure, it is useful to restrict the notion further: a representation of G on the vector space V is a group homomorphism:

ρ : G → GL(V),

here GL(V) consists of the invertible linear transformations of V. To every group element g is assigned an automorphism ρ(g) so that ρ(g) ° ρ(h) = ρ(gh) for any h in G.

This can be understood in 2 ways, both of which give rise to whole new domains of mathematics. It can yield new information about the group G. Representation theory of finite groups and representations of Lie group are 2 main sub domains of the theory. The totality of representations can be governed by the group's characters. 

All Symmetry Operations which are associated with isolated molecules can be characterized as Rotations, which are known as proper and improper rotations.A central construct in inorganic chemistry is theory of the molecular symmetry. Mathematical group theory gives the language to describe the shapes of molecules according to their point group symmetry. Group theory enables factoring and simplification of theoretical calculations. Spectroscopic features are analyzed and described with respect to the symmetry properties of the, vibration or electronic states. Knowledge of the symmetry properties of the ground and excited states allows predicting the numbers and intensities of absorptions in vibration and electronic spectra. A classic application of group theory is prediction of the number of C-O vibrations in substituted in the metal carbonyl complexes.