Definition:Permutation Group
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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.6$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{EE}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 76$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$: Subgroups
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(4) \ \text {(a)}$
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.5$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): permutation group
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): substitution group