Problem 22 - Algebra: Pure and Applied

Question

Consider the relation \(R\) on the class of all groups defined by the condition that \(G R G^{\prime}\) if and only if \(G\) and \(G^{\prime}\) are isomorphic. Show that \(R\) has the properties of an equivalence relation (reflexivity, symmetry, transitivity).

Step-by-Step Solution

Answer

The relation \( R \) is an equivalence relation because it satisfies reflexivity, symmetry, and transitivity.

Step 1: Define Reflexivity

Reflexivity means that for any group \( G \), the statement \( G \, R \, G \) must hold. A group is isomorphic to itself via the identity map. Consequently, every group \( G \) is related to itself, satisfying reflexivity.

Step 2: Define Symmetry

Symmetry requires that if \( G \, R \, G' \), then \( G' \, R \, G \). If two groups \( G \) and \( G' \) are isomorphic, there exists an isomorphism from \( G \) to \( G' \). The inverse of this isomorphism is an isomorphism from \( G' \) to \( G \), ensuring symmetry.

Step 3: Define Transitivity

Transitivity means that if \( G \, R \, G' \) and \( G' \, R \, G'' \), then \( G \, R \, G'' \). If there is an isomorphism from \( G \) to \( G' \) and from \( G' \) to \( G'' \), the composition of these two isomorphisms is an isomorphism from \( G \) to \( G'' \), establishing transitivity.

Key Concepts

Group Isomorphism

In the realm of abstract algebra, group isomorphism is a fundamental concept. It refers to a relationship between two groups, where they can be transformed into each other by a bijective function that preserves the group operation. This means that if you have a group \(G\) and another group \(G'\), an isomorphism allows you to map elements of \(G\) to elements of \(G'\) such that the structure and properties of the groups remain intact.
For example:

• If \( \phi : G \to G'\) is an isomorphism, it ensures that for every pair of elements \(a, b \in G\), \( \phi(ab) = \phi(a)\phi(b)\).
• Both the function and its inverse are bijective, meaning they are one-to-one and onto.

The notion of isomorphism is crucial because it allows mathematicians to consider groups that are structurally the same without having to delve into their specific elements. It’s about understanding the essence of the group rather than its individual parts.

Reflexivity

Reflexivity is one of the defining features of an equivalence relation. It asserts that every element is related to itself. In the context of group isomorphisms, reflexivity implies that any group \(G\) is isomorphic to itself. The simplest isomorphism that does this is the identity map on \(G\).
The identity map has the following properties:

• It maps each element of a group to itself, ensuring the operation of the group is preserved.
• Every element in \(G\) remains unchanged, showing a perfect match from the group to itself.

Therefore, reflexivity in terms of group isomorphism is naturally satisfied. This self-relatedness is critical in establishing the basic condition for an equivalence relation.

Symmetry

Symmetry is another key feature of equivalence relations. It states that if an element \(a\) is related to an element \(b\), then \(b\) is also related to \(a\). Translating this into group theory terms, if group \(G\) is isomorphic to group \(G'\), then \(G'\) must also be isomorphic to \(G\).
This concept relies on the fact that isomorphisms are bijections, and every bijection has an inverse function:

• If there exists an isomorphism \( \phi : G \to G'\), there must also exist an inverse isomorphism \( \phi^{-1} : G' \to G\).
• The inverse function still respects the group operation, \( \phi^{-1}(xy) = \phi^{-1}(x)\phi^{-1}(y) \) for \(x, y \in G'\).

Thus, symmetry in group isomorphism is easily confirmed by the existence of these inverse mappings.

Transitivity

Transitivity is a vital component of establishing equivalence relations. For groups, this means that if a group \(G\) is isomorphic to a group \(G'\), and \(G'\) is isomorphic to \(G''\), then \(G\) should be isomorphic to \(G''\).
This is achieved through the composition of isomorphisms:

• Suppose \( \phi : G \to G'\) and \( \psi : G' \to G''\) are both isomorphisms.
• The composition \( \psi \circ \phi : G \to G''\) combines these functions into a single isomorphism from \(G\) to \(G''\).
• This composed function still retains the properties of a bijection and respects the group structure.

Thus, transitivity is effectively validated, showing how groups maintain their isomorphic relationships across chains of isomorphic mappings.