In mathematics a group \(G := \{S,*\}\) consists of a set \(S\) (such as the the set of all integers \(\mathbb{Z} \)),
and a binary operation \(*\) (such as addition \(+\)) where the following 4 criteria are met:
Closure: For any \(a,b \in G \) we have \( a*b \in G\). So the result of doing the binary operation
on any 2 elements from the group will return an element that is also in the group.
Associativity: For any \(a,b,c \in G \) we have \( (a*b)*c = a*(b*c) \).
Identity: There exists an element \(e \in G\) such that for any \(a \in G\) we have
\( a*e = a\).
Inverse: For any element \(a \in G\) there exists an element \(a^{-1} \in G\) such that
\( a*a^{-1} = e\).
The group \(G := \{\mathbb{Z},+\}\) satisfies all of these conditions where \(e=0\),
\( a^{-1} = -a \), the associative law holds for integers and the sum of two integers is also an integer.
Group theory, particularly group theory for finite groups, has certain useful applications to condensed matter and
quantum physics. One kind of group that is of interest is the permutation group denoted \(S_n\). The group elements consist of
each possible permutation of a set of \(n\) elements. For example consider the permutation group of 3 elements \(S_3\) with
the 3 elements denoted as (123). One element in the group \(S_3\) is the permutation that swaps the first and second element,
\[1 \rightarrow 2 \]
\[2 \rightarrow 1 \]
\[3 \rightarrow 3 \]
so the set (123) becomes (213). What is also interesting about this group is that it is isomorphic to the group
of symmetry operations of an equilateral triangle. This group is denoted as \(D_3\) and is one of the Dihedral groups. The group
consists of 6 elements: 3 2-fold rotations of \(180^\circ\) about 3 different axises each axis bisecting the triangle through
one of its vertices, 2 3-fold rotations of \(120^\circ\) about the axis that comes out of the screen and through the center of
the triangle and the identity symmetry operation which leaves the triangle unchanged. What is important about each of these operations
is that they preserve the orientation of the triangle.
The main idea behind applying group theory to condensed matter physics is that many molecules exhibit these same symmetries
and that by identifying which symmetry group a molecule belongs to we can use theorems from group & representation theory
to discover certain nontrivial properties and behaviors of said molecule. A more advanced example would be how you can use
group and representation theory to figure out how the energy levels of a certain molecule will split when placed in a static electric
field produced by charges placed in a certain configuration around the molecule.
For another example consider a hypothetical molecule that has oxygen atoms placed at each
vertex of an equilateral triangle as shown below with each vertex labeled from 1 to 3.
This molecule can be classified as belonging to the group of \(D_3\) as well as \(S_3\).