Algebraic and Stochastic Coding Theory

B

Some Useful Groups

B.1    Permutation Groups

A group whose elements are permutations on a given finite set of objects (symbols) is called a permutation group on these objects. Let a₁, a₂, …, a_n denote n distinct objects, and let b₁, b₂, …, b_n be any arrangement of the same n objects. The operation of replacing each a_i by b_i, 1 ≤ i ≤ n, is called a permutation performed on the n objects. It is denoted by

$S:\left(\begin{array}{l}{a}_1{a}_2\cdots a_n\\ b_1{b}_2\cdots b_n\end{array}\right)$

(B.1)

and is called a permutation of degree n. If the symbols on both lines in (B.1) are the same, the permutation is called the identical permutation and is denoted by I. Let

$\begin{array}{cc}T:\left(\begin{array}{l}{b}_1{b}_2\cdots b_n\\ c_1{c}_2\cdots c_n\end{array}\right),& U:\left(\begin{array}{l}{a}_1{a}_2\cdots a_n\\ c_1{c}_2\cdots c_n\end{array}\right),\end{array}$

then U is the product of S and T, i.e., U = ST, since

$U=ST=\left(\begin{array}{l}{a}_1{a}_2\cdots a_n\\ b_1{b}_2\cdots b_n\end{array}\right)\left(\begin{array}{l}{b}_1{b}_2\cdots b_n\\ c_1{c}_2\cdots c_n\end{array}\right)=(\begin{array}{l}{a}_1{a}_2\cdots a_n\\ c_1{c}_{}\end{array}$