Example of an elementary permutation

Question

Can someone please give an example of an elementary permutation? The book says that every permutation can be written as a composite of elementary permutations. Can someone please give an example?

Thanks in advance!

Answer 1

Every permutation can be written as a composition of transpositions (as a composition of "elementary permutations").

Such an elementary permutation (transposition) permutes (swaps) two elements of a given set, e.g., consider the transposition $(12) = (21),$ which means $$1\mapsto 2\;\text{ and }\;2\mapsto 1.$$ In essence, a transposition is a $2$-cycle.

Suppose we take a cyclic permutation $(1234) \in S_4$, where $S_4$ is the group of permutations on $\{1, 2, 3, 4\}$. We can express it as the compostion of transpositions: $(1234) = (14)(13)(12).$

Answer 2

I guess this may refer to James Munkres' Analysis on Manifolds.

Chapter 6 Differential forms, Lemma 27.1.

Definition. Given $ 1 \le i < k $, let $e_i$ be the element of $S_k$ symmetric group defined by setting $$\underbrace{e_i\overbrace{(j) = j}^{\text{elements stay in their places}} \quad \text{for} \quad j \ne i, i+1;}_{\text{e.g. for $e_5$ all elements except for j=5 and j=i+1=6 remain the same, but items 5 and 6 are swapped, see rules below}}$$ and

$$ e_i(i)=i+1 \quad \text{and} \quad e_i(i+1)=i$$

We call $e_i$ and elementary permutation.

In other words, elementary permutation is a swap of two adjacent elements, all other elements in this permutation remain the same.