Group Theory

Definition

A group $G$ may be defined as the set of objects, called the elements $g$ of $G$ that may be combined to form a well defined product in $G$ that satisfies the following conditions

  1. If $a$ and $b$ are any two elements in $G$ then $ab$ is also an element in G.
  2. If $a$, $b$ and $c$ are three elements in G, then multiplication is associative: $a(bc) = (ab)c$.
  3. There is a unit or identity element in $G$ such that for every $a \in G$, $1a = a1 = a$.
  4. There exists an inverse or reciprocal element $a^{-1}$ for every element $a$ in $G$ such that $aa^{-1} = a^{-1}a = 1$.

It is clear from the definition that the identity and inverse elements are both unique. Suppose there are two identity elements, namely 1 and 1'. Then $11' = 1 = 1'1 = 1'$. Suppose there are two inverse elements of $a \in G$, namely $a^{-1}$ and $a'^{-1}$. Then, we have $aa^{-1} = a^{-1}a = 1 = aa'^{-1}$. Premultiplying both sides by $a^{-1}$ we get $a^{-1} = a'^{-1}$.

It should be noted here that the elements of a group are in general not elements of a scalar field, and hence multiplicative commutativity does not hold in general for them. For now, you can think of the elements of a group as matrices, an analogy that we will use soon.

Interpretation in terms of Symmetry Operations

With the elements of a group $G$, we may associate symmetry operations. Hence, the elements correspond to generally distinct symmetry operations. The first condition stated above implies that if the symmetry operation $b$ is carried out on a system before the symmetry operation $a$ then, $ab$ is equivalent to a single symmetry operation in $G$. In fact, symmetry operations are described using group theory, a firm mathematical theory to deal with invariants and symmetries.

Rotation Matrices

In $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$, the rotation of a coordinate system is specified by a rotation angle and the transformation in terms of a matrix known as the rotation matrix. For instance, consider the transformation

(1)
\begin{align} \left[\begin{array}{cc}x'\\y'\end{array}\right] = \left[\begin{array}{cc}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{array}\right]\left[\begin{array}{cc}x\\y\end{array}\right] = R_{\theta}\left[\begin{array}{cc}x\\y\end{array}\right] \end{align}

This describes a rotation of the $(x,y)$ space into the $(x',y')$ space through an angle $\theta$. The rotation matrix $R_{\theta}$ is an element of a rotation group called SO(2), which stands for Special Orthogonal Rotations in 2 Dimensions. It is easily seen that

(2)
\begin{align} R_{\theta_1}R_{\theta_2} = R_{\theta_2}R_{\theta_1} \end{align}

that is, the rotation operations commute and hence the group is said to be Abelian. The inverse of every rotation $\theta$ is $-\theta$ and the identity element is the 2x2 identity matrix. Note that no mention has been made of the third dimension. In fact, since $\mathbb{R}^2 \subset \mathbb{R}^{3}$, we also have $SO(2) \subset SO(3)$ where $SO(3)$ denotes the Special Orthongal Rotations in 3 dimensions. So, the 'full' rotation matrix for the above rotation transformation, given by

(3)
\begin{align} \left[\begin{array}{cc}x'\\y'\\z'\end{array}\right] = \left[\begin{array}{ccc}\cos\theta&\sin\theta&0\\-\sin\theta&\cos\theta&0\\0&0&1\end{array}\right]\left[\begin{array}{cc}x\\y\\z\end{array}\right] = R_{\theta}\left[\begin{array}{cc}x\\y\\z\end{array}\right] \end{align}

is an element of $SO(3)$. So we can identify rotations in Euclidean space as operations in the $SO(3)$ group. The term "special" is used because the determinant of these rotation matrices is +1 and the angle $\theta$ varies continuously from $0$ to $2\pi$. The term orthogonal is used as the rotation matrices are all orthogonal. This also means that the transpose of the rotation matrix is its inverse [as the transpose of a matrix always exists, the inverse of every rotation matrix exists, consisent with our earlier observation that $R_{\theta}^{-1} = R_{-\theta}$. Further, $SO(3)$ is an Abelian group, as stated in equation (2).

Note that rotation matrices describing rotations about the x and y axes are also members of $SO(3)$ and so are matrices about any fixed line (treated as an axis of rotation) in $\mathbb{R}^{3}$. The example considered above has referred to only a special case of rotation about the z axis in $\mathbb{R}^{3}$.


Subgroup

A subgroup G' of a group G is a group consisting of elements of G such that the product of any two elements of G' is again in the subgroup G' (the concept of a subgroup will be familiar as a subspace to those who have independently studied linear algebra). G' is said to be closed under the multiplication of G. As an example, the identity element 1 of G forms a subgroup of G. The unit transformation ($\theta = 0$) and the rotation with $\theta = \pi$ about some axis form a finite subgroup of the group of rotations about that axis.

Invariant Subgroup

If $gg'g^{-1}$ is an element of G' for any $g \in G$ and $g' \in G'$, then G' is called an invariant subgroup of G. If the elements are matrices, then this element corresponds to a similarity transformation of $g'$ in G' by an element $g$ of G. The identity element forms an invariant subgroup of G because $g1g^{-1} = 1$.


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