Permutation and Levi-Civita Symbol

Permutation

Definition 1. A permutation of a set $A$ is a function $\phi : A \rightarrow A$ that is bijective. 

Remark. Let denote the function composition $\sigma \circ \tau$ for permutations $\sigma$, $\tau$ by $\sigma \tau$. Then $\sigma \tau$ is bijective.

임의의 자연수 $n$에 대해 집합 $A = \{ 1, 2, ..., n \}$가 주어져 있다. 이때 $A$의 permutation, 즉 치환은 각 자연수를 다른 자연수로 보내는 작용을 하는 함수이며, 한마디로 순서를 바꿔주는 함수이다. 

Symmetric Group

Definition 2. Let $A$ be the finite set $\{ 1, 2, ..., n\}$. The group of all permutations of $A$ is the symmetric group on $n$ letters, and is denoted by $S_n$.

Definition 3. We define the equivalence relation $\sim$ in $A$ if the following condition holds: 
For $a, b \in A$, let $a \sim b$ $\iff$ $b = \sigma^n(a)$ for some $n \in \mathbb{Z}$.

위와 같이 relation을 정의하면 equivalence relation이 됨은 쉽게 보일 수 있다. $A$의 원소에 동일한 permutation을 유한 번 적용하여서 어떤 원소를 얻어낼 수 있다면, 두 원소는 위에서 정의한 관계에 있다고 말한다.

Orbit

Definition 4. Let $\sigma$ be a permutation of a set $A$. The equivalence classes in $A$ determined by the equivalence relation defined above are the orbits of $\sigma$.

Cycle

Definition 5. A permutation $\sigma \in S_n$ is a cycle if it has at most one orbit containing more than one element. The length of a cycle is the number of elements in its largest orbit.

Transposition

Definition 6. A cycle of length $2$ is a transposition.

Parity

Definition 7. A permutation of a finite set is even or odd according to whether it can be expressed as a product of an even number of transpositions or an odd number of transpositions, respectively. 

Levi-Civita Symbol

Definition 8. The Levi-Civita symbol is defined by: $$\varepsilon_{ij...} = \begin{cases} +1 & \text{if } (i, j, ...) \text{ is an even permutation of } (1, 2, ...) \\ -1 & \text{if } (i, j, ...) \text{ is an odd permutation of } (1, 2, ...) \\ 0 & \text{otherwise} \end{cases}$$

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Reference is here: https://product.kyobobook.co.kr/detail/S000022899747

First Course in Abstract Algebra | John Fraleigh - 교보문고