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:

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

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 $a\neq b$ 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 $a=5$:

$\sigma$ in cycle decomposition	$\sigma '$ in cycle decomposition
$(1,2,3)$	$(1,2,3)$
$(1,2,3,5)$	$(1,2,3)$
$(1,4)(2,5)$	$(1,4)$
$(1,2)(3,4,5)$	$(1,2)(3,4)$

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.