Cyclic permutation of three elements in a list:

Equivalent action with a permutation list:

Take the lowercase alphabet:

Exchange the first and last character:

Permute several characters:

Permute an expression under all elements of a group:

Permute the parts of an expression:

Permute the parts of an expression under all elements of a group:

Give a permutation in list form. The length of the expression does not change:

The eight possible rotations and reflections of a square:

Permute never changes the number of parts of an expression. It simply reorders them:

However, Part can change the number of parts:

When applied to the identity permutation list, Permute is the inverse of PermutationReplace:

Another way of inverting the action of Permute is using FindPermutation:

When all parts of the expression are different, the permutation can be uniquely recovered:

Permute is a right action with respect to the product of permutations:

Dimensions[Transpose[array,perm]] is equivalent to Permute[Dimensions[array],perm]:

Compute the same results with the Cycles form of the permutation:

Applying Permute to a vector is equivalent to multiplying the vector from the right with a permutation matrix:

Permute and Part interpret permutation lists in different ways:

Permute places the first object at the third position, the second object at the first position and the third object at the second position:

Part places the third object at the first position, the first object at the second position and the second object at the third position:

Those interpretations are effectively the inverse of each other: