Re: A beginner's question on permutation groups-

Edward Green wrote:
Let H be a set containing n distinct elements.

Let G be the set of all possible permutations of the n elements. The
elements of G form a group whose group operation is the composition of
permutation.

What about the underlying set H? Do we have a general name for it, or
role in the theory?

The underlying set has no special name.

For any pair of sets A, B of n elements there exists a bijection, i.e. a one-to-one mapping from the entire set A onto the entire set B. Its inverse is a bijection from B onto A.
Sometimes it is convenient to speak of a =bijection between= sets A and B; but please keep in mind that this means the one-to-one mapping and its inverse taken together.

Now about the permutation groups! Let P(A), P(B) the groups of all permutations of elements of A and of elements of B.

Apply any bijection from A onto B. Then you see: to any permutation of elements of A there is a corresponding permutation of elements of B.
Consider any composition (*) of two permutations P1 and P2 on set A. There are corresponding permutations Q1, Q2 on set B, and the permutation corresponding to the product permutation P1.P2 is the product permutation Q1.Q2.
This argumentation also works from B back to A. This is because of the bijection from B onto A, which is the inverse mapping of the bijection from A onto B.

In the end one observes that the permutation groups P(A) and P(B) are isomorphic, i.e. they have the same structure, the same composition table. So when studying "the" full permutation group on n elements in its own right the underlying set is no longer considered. Only the number of its elements is important. Therefore there is no special name for the underlying set.

Perhaps there is a simpler way of explaining all this once one has observed that a permutation on a set A is by definition a bijection of A onto itself.


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(*)
Composition of group elements is commonly denoted as group multiplication. Presumably this term stems from the group properties of common fractions under multiplication as their group composition law.
In commutative groups the group composition is often denoted as addition, in analogy of addition in ordinary arithmetic.
Why "addition" for commutative groups and "multiplication" for non-commutative groups?
Presumably because there exist non-commutative algebras (for instance the quaternions),
where the one composition, the addition, is always commutative, while the other composition, the multiplication, need not be commutative.


Happy studies: Johan E. Mebius
.

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