# 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.