Definition:Permutation on n Letters/Two-Row Notation

Definition

Let $\pi$ be a permutation on $n$ letters.

The two-row notation for $\pi$ is written as two rows of elements of $\N^*_n$, as follows:

$\pi = \begin{bmatrix} 1 & 2 & 3 & \ldots & n \\ \pi \left({1}\right) & \pi \left({2}\right) & \pi \left({3}\right) & \ldots & \pi \left({n}\right) \end{bmatrix}$

The bottom row contains the effect of $\pi$ on the corresponding entries in the top row.

Alternative Notation

Some sources use round brackets for the two-row notation:

$\pi = \begin{pmatrix} 1 & 2 & 3 & \ldots & n \\ \pi \left({1}\right) & \pi \left({2}\right) & \pi \left({3}\right) & \ldots & \pi \left({n}\right) \end{pmatrix}$

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 78$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (4)$