Definition:Permutation Group

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Definition

A permutation group on a set $S$ is a subgroup of the symmetric group $\struct {\map \Gamma S, \circ}$ on $S$.


Also known as

Some sources call this a group of permutations, but this can easily be confused with the group of permutations (that is, the Symmetric Group itself).

A permutation group is sometimes referred to as a concrete group, based on the idea that it is a specific instantiation of a group which can be perceived as such in its own right, as opposed to an abstract group which consists purely of a set with an abstractly defined operation.


Some sources use the name substitution group.


Other sources use the name transformation group.

However, the term transformation group is also encountered in context of group actions.

Hence its use in this context is discouraged so as to avoid confusion.


Examples

Example on $\R$

Let $S = \R_{\ge 0} \times \R$ denote the Cartesian product of $\R_{\ge 0}$ and $\R$.


Let $\tuple {a, b} \in S$.

Let $f_{a, b}: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map {f_{a, b} } x := a x + b$

Let $\GG$ be the set defined as:

$\GG = \set {f_{a, b}: \tuple {a, b} \in S}$


Let $\struct {S, \oplus}$ be the group where $\oplus$ is defined as:

$\forall \tuple {a, b}, \tuple {c, d} \in S: \tuple {a, b} \oplus \tuple {c, d} := \tuple {a c, a d + b}$


Then $\struct {\GG, \circ}$ is a permutation group on $\R$ which is isomorphic to $\struct {S, \oplus}$.


Also see

  • Results about permutation groups can be found here.


Sources

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