# 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**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 8$ - 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): Entry:**permutation group** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**substitution group**