Derived permutation

Definition
Suppose $$S$$ is a set, $$a \in S$$ is an element, and $$T = S \setminus \{ a \}$$. Let $$\sigma$$ be a permutation on $$S$$. The derived permutation from $$\sigma$$ on $$T$$ is the permutation $$\sigma'$$ of $$T$$ defined as follows: consider the cycle decomposition of $$\sigma$$. Take the cycle containing $$a$$ and remove $$a$$ from it, to get a permutation $$\sigma'$$ on $$T$$. Explicitly:

$$\sigma'(b) = \left \lbrace\begin{array}{rr} \sigma(b), & \qquad b \ne \sigma^{-1}(a) \\ \sigma(a), & \qquad b = \sigma^{-1}(a)\end{array}\right.$$

Note that if $$a$$ is a fixed point of $$\sigma$$, the derived permutation is simply the restriction of $$\sigma$$ to $$T$$.

The term is typically used when $$S = \{ 1,2,\dots, n-1,n \}$$, $$a = n$$, and $$T = \{ 1,2,\dots, n-1 \}$$.

Facts

 * When $$S$$ has size $$1$$ or $$2$$, the derived permutation mapping is a homomorphism. For size greater than $$2$$, it is not a homomorphism. However, its restriction to the isotropy subgroup of $$\{ a \}$$ is an isomorphism.
 * The derived permutation mapping preserves inverses.
 * Suppose $$a \ne b$$ are elements of $$S$$. Then, the composite of the derived permutation mappings $$\operatorname{Sym}(S)$$ to $$\operatorname{Sym}(S \setminus \{ a \})$$ to $$\operatorname{Sym}(S \setminus \{ a,b \})$$ is the same as the composite $$\operatorname{Sym}(S)$$ to $$\operatorname{Sym}(S \setminus \{ b \})$$ to $$\operatorname{Sym}(S \setminus \{ a,b \})$$.

Examples
Here, $$S = \{ 1,2,3,4,5 \}$$ and $$a = 5$$:

Related notions

 * Virtual permutation is a sequence of permutations on sets $$\{ 1,2, \dots, n \}$$, with $$n \to \infty$$, with each permutation the derived permutation of the next.