Problem 1 - A Book of Abstract Algebra

Question

The symbol Aut \((G)\) is used to designate the set of all the automorphisms of \(G\). Prove that the set Aut \((G)\), with the operation o of composition, is a group by proving that \(\operatorname{Aut}(G)\) is a subgroup of \(S_{G}\).

Step-by-Step Solution

Answer

Aut \((G)\) is a subgroup of \(S_{G}\) because it is closed under composition, contains the identity, and each element has an inverse in the set.

Step 1: Define an Automorphism

An automorphism is a bijective homomorphism from a group \( G \) to itself. This means it is a function \( f: G \to G \) that satisfies two properties: it is bijective (one-to-one and onto) and respects the group operation, i.e., for all \( a, b \in G \), \( f(ab) = f(a)f(b) \).

Step 2: Explain the Symmetric Group

The symmetric group \( S_G \) consists of all bijective functions from \( G \) to \( G \) with the operation of function composition. Thus, any automorphism of \( G \) is a member of \( S_G \) because it is bijective.

Step 3: Show Closure under Composition

To show \( \operatorname{Aut}(G) \) is a group, we first show closure: if \( f, g \in \operatorname{Aut}(G) \), then \( f \circ g \) is also an automorphism. Since \( f \) and \( g \) are bijective homomorphisms, their composition \( f \circ g \) is also bijective, and for any \( a, b \in G \), \[ (f \circ g)(ab) = f(g(ab)) = f(g(a)g(b)) = f(g(a))f(g(b)) = (f \circ g)(a)(f \circ g)(b). \] Thus, \( f \circ g \) is a homomorphism, showing closure.

Step 4: Identify Identity Element

The identity automorphism \( \text{id}: G \to G \) defined by \( \text{id}(a) = a \) for all \( a \in G \) is in \( \operatorname{Aut}(G) \). Since \( \text{id} \) is bijective and respects the group operation, it acts as the identity element in \( \operatorname{Aut}(G) \) under composition.

Step 5: Prove Existence of Inverses

For each automorphism \( f \in \operatorname{Aut}(G) \), its inverse \( f^{-1} \) exists since \( f \) is bijective. The inverse is also an automorphism because for all \( a, b \in G \), \[ f(f^{-1}(a)f^{-1}(b)) = f(f^{-1}(a))f(f^{-1}(b)) = ab, \] ensuring that \( f^{-1}(ab) = f^{-1}(a)f^{-1}(b) \), making \( f^{-1} \) a homomorphism.

Step 6: Conclude is a Subgroup of \( S_G \)

Since \( \operatorname{Aut}(G) \) is shown to be closed under composition, contains the identity element, and every element has an inverse in the set, it satisfies the group axioms. We have established that \( \operatorname{Aut}(G) \) is a subgroup of \( S_G \) since all its elements are bijective functions on \( G \).

Key Concepts

Automorphism

An automorphism is a specific type of function within the realm of group theory. This concept is crucial in understanding the internal symmetries of a group. An automorphism is a bijective homomorphism from a group \( G \) to itself. This means that it is not only a mapping of the elements in the group but does so in a way that preserves the group's structure.

• **Bijective**: Each element in the group maps to a unique element, leading to both one-to-one and onto relationships.
• **Homomorphism**: This ensures the function respects the group operation, so for any elements \( a \) and \( b \) in \( G \), the equation \( f(ab) = f(a)f(b) \) holds true.

Automorphisms play a vital role in determining the nature and structure of a group, often revealing the inherent symmetries and properties that are not immediately visible from the group's elements alone.

Symmetric Group

The symmetric group \( S_G \) consists of all possible permutations of a set, which in our context is the group \( G \). These permutations take the form of bijective functions from the set \( G \) to itself, allowing us to rearrange the elements of \( G \) in every possible way.

• **Bijective Functions**: Each function is a rearrangement of the group elements, ensuring that no two elements are mapped to the same element.
• **Operation of Function Composition**: When these functions are combined via composition, the resulting function is also bijective, making \( S_G \) a group under composition.

Understanding \( S_G \) is key to recognizing how elements can be rearranged systematically, and why automorphisms, as bijective functions of \( G \), naturally fit within this group.

Group Homomorphism

A group homomorphism is a function between two groups that respects the underlying operations of those groups. In simpler terms, it shows how the structure or the operation of one group can be mirrored or translated into another group.
For automorphisms, since the group in question maps onto itself, the homomorphism condition asserts that the operation within \( G \) is preserved. Hence for any \( a, b \in G \), we see that \( f(ab) = f(a)f(b) \).

• Ensures that the operation of combining elements within the group is reflected accurately through the function.

The concept of homomorphisms provides a connective link, showing how different groups can relate, allowing complex structures to be broken down into more manageable parts.

Bijective Function

A bijective function is central to understanding automorphisms and many concepts in group theory. It is a function that pairs every element from one set with a unique element from another set, in this case, the same group \( G \).

• **One-to-one (Injective)**: No two elements are mapped to the same element, ensuring distinctness.
• **Onto (Surjective)**: Every element in the group appears as an image, ensuring completeness.

When both these properties are satisfied, the function has an inverse because there’s a clear and direct mapping between elements. This quality is what makes bijective functions pivotal in group theory, as they allow functions to be reversed and further manipulated without losing structure or integrity.