Permutation of a set

Stub grade: A*

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Very important to get some work on the symmetric group in play, then this may be demoted. Demote to grade A once the notation section has been added, there's a lot to say there.

Note: permutation on a set redirects here.

Definition

Let $X$ be any non-empty set, $X$. A permutation on $X$ is:

• A bijective function, $f:X\rightarrow X$. Recall that bijective means injective (1:1) and surjective (onto).

Claims:

The collection of all permutations of a set forms a group under function composition - see the permutation group. The symmetric group is a special case of the permutation group when the set is finite.