# 1.2: Operator Properties and Mathematical Groups

- Page ID
- 221670

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 C_{4}(z) and σ(xz) commute?

… or analyzing with matrix representations,

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

C_{4}(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}_{ }⋅ C_{4}(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 C_{3} and σ_{v}. These do not constitute a group because identity criterion is not satisfied. Do E, C_{3}, σ_{v} form a group? To address this question, a stereographic projection (featuring critical operators) will be used:

So how about closure?

C_{3 }⋅ C_{3} = C_{3} ^{2} (so C_{3} ^{2} needs to be included as part of the group)

Thus E, C_{3} and σ_{v} are not closed and consequently these operators do not form a group. Is the addition of C_{3} ^{2} and σ_{v}´ sufficient to define a group? In other terms, are there any other operators that are generated by C_{3} and σ_{v}?

… the proper rotation axis, C_{3}:

\(C_{3}\)

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

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

\(C _{3} \cdot C _{3} \cdot C _{3} \cdot C _{3}= E \cdot C _{3}= C _{3}\)

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 C_{3} and σ_{v}. Have already seen that C_{3} ⋅ σ_{v} = σ_{v}’ … how about σ_{v} ⋅ C_{3}

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 C_{3} 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: C_{3} → E, C_{3}, C_{3} ^{2}

As mentioned above, E, C_{3}, and C_{3}^{2} meet the conditions of a group; they form a cyclic group. Moreover these three operators are a **subgroup** of E, C_{3}, C_{3} ^{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, C_{3}, C_{3} ^{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

\(E ^{-1} \cdot C _{3} \cdot E = E \cdot C _{3} \cdot E = C _{3}\)

\(C _{3}^{-1} \cdot C _{3} \cdot C _{3}= C _{3}^{2} \cdot C _{3} \cdot C _{3}= E \cdot C _{3}= C _{3}\)

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

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

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

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

∴ C_{3} and C_{3} ^{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.