1.2: Operator Properties and Mathematical Groups

The inverse of A (defined as (A)^–1) is B if A ⋅ B = E

For each of the five symmetry operations:

\(( E )^{-1}= E \Longrightarrow( E )^{-1} \cdot E = E \cdot E = E\)
\((\sigma)^{-1}=\sigma \Longrightarrow(\sigma)^{-1} \cdot \sigma=\sigma \cdot \sigma= E\)
\((i)^{-1}=i \Longrightarrow(i)^{-1} \cdot i=i \cdot i=E\)
\(\left(C_{n}^{m}\right)^{-1}=C_{n}^{n-m} \Longrightarrow\left(C_{n}^{m}\right)^{-1} \cdot C_{n}^{m}=C_{n}^{n-m} \cdot C_{n}^{m}=C_{n}^{n}=E\)
e.g. \(\left(C_{5}^{2}\right)^{-1}=C_{5}^{3}\) since \(C_{5}^{2} \cdot C_{5}^{3}=E\)
\(\left(S_{n}^{m}\right)^{-1}=S_{n}^{n-m}(n \text { even }) \Longrightarrow\left(S_{n}^{m}\right)^{-1} \cdot S_{n}^{m}=S_{n}^{n-m} \cdot S_{n}^{m}=S_{n}^{n}=C_{n}^{n} \cdot \sigma_{h}^{n}=E\)
\(\left(S_{n}^{m}\right)^{-1}=S_{n}^{2 n-m}(n \text { odd }) \Longrightarrow\left(S_{n}^{m}\right)^{-1} \cdot S_{n}^{m}=S_{n}^{2 n-m} \cdot S_{n}^{m}=S_{n}^{2 n}=C_{n}^{2 n} \cdot \sigma_{h}^{2 n}=E\)

Two operators commute when A ⋅ B = B ⋅ A

Example: Do C₄(z) and σ(xz) commute?

… or analyzing with matrix representations,

\(\left[\begin{array}{rrr}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right] \cdot\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{rrr}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]\)

C₄(z) ⋅ σ_xz = σ_d´

Now applying the operations in the inverse order,

… or analyzing with matrix representations,

\(\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{array}\right] \cdot\left[\begin{array}{rrr}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]\)

σ_xz ⋅ C₄(z) = σ_d

\begin{equation}
\therefore \quad C_{4}(z) \sigma(x z)=\sigma_{d}^{\prime} \neq \sigma(x z) C_{4}(z)=\sigma_{d} \Rightarrow \text { so } C_{4}(z) \text { does not commute with } \sigma(x z)
\end{equation}

A collection of operations are a mathematical group when the following conditions are met:

• closure: all binary products must be members of the group
• identity: a group must contain the identity operator
• inverse: every operator must have an inverse
• associativity: associative law of multiplication must hold \[(A ⋅ B) ⋅ C = A ⋅ (B ⋅ C\]

(note: commutation not required… groups in which all operators do commute are called Abelian)

Consider the operators C₃ and σ_v. These do not constitute a group because identity criterion is not satisfied. Do E, C₃, σ_v form a group? To address this question, a stereographic projection (featuring critical operators) will be used:

So how about closure?

C₃ ⋅ C₃ = C₃ ² (so C₃ ² needs to be included as part of the group)

Thus E, C₃ and σ_v are not closed and consequently these operators do not form a group. Is the addition of C₃ ² and σ_v´ sufficient to define a group? In other terms, are there any other operators that are generated by C₃ and σ_v?

… the proper rotation axis, C₃:

\(C_{3}\)
\(C _{3} \cdot C _{3}= C _{3}^{2}\)
\(C _{3} \cdot C _{3} \cdot C _{3}= C _{3}^{2} \cdot C _{3}= C _{3} \cdot C _{3}^{2}= E\)
\(C _{3} \cdot C _{3} \cdot C _{3} \cdot C _{3}= E \cdot C _{3}= C _{3}\)

etc.

\(\therefore C _{3}\) is the generator of \(E , C _{3}\) and \(C _{3}^{2}\), note: these three operators form a group

… for the plane of reflection, σ_v

_{\(\sigma_{v}\)
\(\sigma_{v} \cdot \sigma_{v}=E\)
\(\sigma_{v} \cdot \sigma_{v} \cdot \sigma_{v}=E \cdot \sigma_{v}=\sigma_{v}\)}

etc.
So we obtain no new information here. But there is more information to be gained upon considering C₃ and σ_v. Have already seen that C₃ ⋅ σ_v = σ_v’ … how about σ_v ⋅ C₃

Will discover that no new operators may be generated. Moreover one finds

\(\begin{array}{ccccccc} & E ^{-1} & C _{3}^{-1} & \left( C _{3}^{2}\right)^{-1} & \sigma_{ v }^{-1} & \left(\sigma_{ v }^{\prime}\right)^{-1} & \left(\sigma_{ v }^{\prime \prime}\right)^{-1} \\ \text {inverses } & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ & E & C _{3}^{2} & C _{3} & \sigma_{ v } & \sigma_{ v }^{\prime} & \sigma_{ v }^{\prime \prime}\end{array}\)

The above group is closed, i.e. it contains the identity operator and meets inverse and associativity conditions. Thus the above set of operators constitutes a mathematical group (note that the group is not Abelian).

Some definitions:

Operators C₃ and σ_v are called generators for the group since every element of the group can be expressed as a product of these operators (and their inverses).

The order of the group, designated h, is the number of elements. In the above example, h = 6.

Groups defined by a single generator are called cyclic groups.

Example: C₃ → E, C₃, C₃ ²

As mentioned above, E, C₃, and C₃² meet the conditions of a group; they form a cyclic group. Moreover these three operators are a subgroup of E, C₃, C₃ ², σ_v, σ_v’,σ_v”. The order of a subgroup must be a divisor of the order of its parent group. (Example h_subgroup = 3, h_group = 6 … a divisor of 2.)

A similarity transformation is defined as: v ^-1 ⋅ A ⋅ ν = B where B is designated the similarity transform of A by x and A and B are conjugates of each other. A complete set of operators that are conjugates to one another is called a class of the group.

Let’s determine the classes of the group defined by E, C₃, C₃ ² , σ_v, σ_v’,σ_v”… the analysis is facilitated by the construction of a multiplication table

\[\begin{array}{l|llllll}
& E & C _{3} & C _{3}^{2} & \sigma_{ v } & \sigma_{ v }^{\prime} & \sigma_{ v }^{\prime \prime} \\
\hline E & E & C _{3} & C _{3}^{2} & \sigma_{ v } & \sigma_{ v }^{\prime} & \sigma_{ v }^{\prime \prime} \\
C _{3} & C _{3} & C _{3}^{2} & E & \sigma_{ v }^{\prime} & \sigma_{ v }^{\prime \prime} & \sigma_{ v } \\
C _{3}^{2} & C _{3}^{2} & E & C _{3} & \sigma_{ v }^{\prime \prime} & \sigma_{ v } & \sigma_{ v }^{\prime} \\
\sigma_{ v } & \sigma_{ v } & \sigma_{ v }^{\prime \prime} & \sigma_{ v }^{\prime} & E & C _{3}^{2} & C _{3} \\
\sigma_{ v }^{\prime} & \sigma_{ v }^{\prime} & \sigma_{ v } & \sigma_{ v }^{\prime \prime} & C _{3} & E & C _{3}^{2} \\
\sigma_{ v }^{\prime \prime} & \sigma_{ v }^{\prime \prime} & \sigma_{ v }^{\prime} & \sigma_{ v } & C _{3}^{2} & C _{3} & E
\end{array}\]

may construct easily using stereographic projections

\(E ^{-1} \cdot C _{3} \cdot E = E \cdot C _{3} \cdot E = C _{3}\)
\(C _{3}^{-1} \cdot C _{3} \cdot C _{3}= C _{3}^{2} \cdot C _{3} \cdot C _{3}= E \cdot C _{3}= C _{3}\)
\(\left( C _{3}^{2}\right)^{-1} \cdot C _{3} \cdot C _{3}^{2}= C _{3} \cdot C _{3} \cdot C _{3}^{2}= C _{3} \cdot E = C _{3}\)
\(\sigma _{ v }^{-1} \cdot C _{3} \cdot \sigma_{ v }=\sigma_{ v } \cdot C _{3} \cdot \sigma_{ v }=\sigma_{ v } \cdot \sigma_{ v }^{\prime}= C _{3}^{2}\)
\(\left(\sigma_{ v }^{\prime}\right)^{-1} \cdot C _{3} \cdot \sigma_{ v }^{\prime}=\sigma_{ v }^{\prime} \cdot C _{3} \cdot \sigma_{ v }^{\prime}=\sigma_{ v }^{\prime} \cdot \sigma_{ v }^{\prime \prime}= C _{3}^{2}\)
\(\left(\sigma_{ v }^{\prime \prime}\right)^{-1} \cdot C _{3} \cdot \sigma_{ v }^{\prime \prime}=\sigma_{ v }^{\prime \prime} \cdot C _{3} \cdot \sigma_{ v }^{\prime \prime}=\sigma_{ v }^{\prime \prime} \cdot \sigma_{ v }= C _{3}^{2}\)

∴ C₃ and C₃ ² from a class

Performing a similar analysis on σ_v will reveal that σ_v, σ_v’ and σ_v’’ form a class and E is in a class by itself. Thus there are three classes:

\(E ,\left( C _{3}, C _{3}^{2}\right),\left(\sigma_{ v }, \sigma_{ v }^{\prime}, \sigma_{ v }^{\prime \prime}\right)\)

Additional properties of transforms and classes are:

• no operator occurs in more than one class
• order of all classes must be integral factors of the group’s order
• in an Abelian group, each operator is in a class by itself.