The Fundamental Reason Why the Set of All Possible Permutations over N Objects is Not a Ring

In mathematics, a ring is a set equipped with two binary operations satisfying a set of axioms. Understanding this concept is fundamental in abstract algebra. In this article, we will explore why the set of all possible permutations over N objects, along with the permutation operator, does not form a ring. We will delve into the necessary operations and properties required for a set to be considered a ring and explain why the permutations lack the required property of addition.

Introduction to Rings

A ring is a set R together with two binary operations, usually denoted as (addition) and * (multiplication). These operations must satisfy the following axioms:

Closure: The sum or product of any two elements in the set is also in the set. Associativity: Addition and multiplication are associative operations, i.e., (a b) c a (b c) and (a * b) * c a * (b * c). Commutativity: Addition is commutative, i.e., a b b a. (Multiplication may or may not be commutative.) Identity element: There exists an additive identity 0 such that a 0 a for all a in the set, and a multiplicative identity 1 such that a * 1 a. Additive inverses: For every element a, there exists an element -a such that a (-a) 0.

In the context of permutations, the multiplication operation is the composition of permutations. However, finding a natural way to define an addition operation that satisfies all the ring axioms is challenging.

The Permutation Group

The permutation group SN is the symmetric group on N elements. The elements of SN are all possible permutations of N objects, and the binary operation is the composition of permutations. Composition of permutations is well-defined, associative, and has an identity element, the identity permutation, which leaves every object fixed. However, defining an addition operation on SN is not straightforward.

Why Addition is Missing

For SN to be a ring, it must include an addition operation that satisfies the ring axioms. Let's explore why this is not possible:

Closure Property

The closure property requires that the sum of any two elements in the set is also in the set. However, there is no natural way to define an addition operation for permutations that results in a permutation. For example, if we take two permutations σ and τ, and try to add them, there is no clear intuitive way to define what this sum would mean or how it would result in another permutation.

Commutativity of Addition

For addition to be commutative, the sum of σ and τ must be the same as the sum of τ and σ. However, there is no natural operation that would make this claim true for all permutations.

Additive Inverses

In a ring, every element must have an additive inverse, i.e., an element that, when added to it, results in the additive identity. For permutations, it is not clear how to define an inverse that leaves the set of permutations closed under addition. For instance, if we take an identity permutation, there is no permutation that, when composed with the identity, results in the identity permutation in the usual sense of addition.

Conclusion

Thus, the set of all possible permutations over N objects along with the permutation operator does not form a ring because the natural definition of addition for permutations is not available. The absence of a suitable addition operation that satisfies all the ring axioms is the primary reason. The permutation group SN under composition is well-defined and has a natural identity element. However, it lacks the required additive structure to be a ring.

Understanding this concept is crucial for those delving into abstract algebra and the structure of groups and rings. The study of permutation groups, like SN, is an important foundation in this field.