Problem 37 - A First Course in Abstract Algebra

Question

a. Show that all automorphisms of a group \(G\) form a group under function composition. b. Show that the inner automorphisms of a group \(G\) form a normal subgroup of the group of all automocptrisms of \(G\) under function composition. [Waming: Be sure to show that the inner automorphisms do form a subgroup.]

Step-by-Step Solution

Answer

Automorphisms of a group form a group under composition. Inner automorphisms are a normal subgroup.

Step 1: Define Automorphisms of a Group

An automorphism is an isomorphism from a group to itself. This means it is a bijective homomorphism, which preserves the group operation. Let \( \text{Aut}(G) \) denote the set of all automorphisms of a group \( G \).

Step 2: Show Automorphisms Form a Group

To show that \( \text{Aut}(G) \) forms a group, we need to show closure, associativity, identity, and inverses under function composition. 1. **Closure:** If \( f, g \in \text{Aut}(G) \), then \( f \circ g \) is an automorphism since the composition of two bijections is bijective, and the homomorphism property is preserved.2. **Associativity:** Function composition is associative.3. **Identity:** The identity map \( \text{id}_G \) is an automorphism and serves as the identity element.4. **Inverses:** For every \( f \in \text{Aut}(G) \), its inverse \( f^{-1} \) is also an automorphism, being both bijective and homomorphic.

Step 3: Define Inner Automorphisms

An inner automorphism is a map \( \phi_g: G \to G \) defined by \( \phi_g(x) = gxg^{-1} \) for a fixed \( g \in G \). The set of all such maps is denoted by \( \text{Inn}(G) \).

Step 4: Show Inner Automorphisms Form a Subgroup

To show \( \text{Inn}(G) \) is a subgroup of \( \text{Aut}(G) \), check:1. **Closure:** If \( \phi_g, \phi_h \in \text{Inn}(G) \), then \( \phi_g \circ \phi_h = \phi_{gh} \) is also in \( \text{Inn}(G) \).2. **Identity:** The map \( \phi_e \), where \( e \) is the identity element of \( G \), is the identity automorphism.3. **Inverses:** For \( \phi_g \in \text{Inn}(G) \), \( \phi_{g^{-1}} \) is its inverse.

Step 5: Show Inner Automorphisms Form a Normal Subgroup

For a subgroup \( N \) to be normal in \( H \), for every \( h \in H \) and \( n \in N \), the element \( hnh^{-1} \) must also be in \( N \). Given \( n \in \text{Inn}(G) \, \text{and} \, h \in \text{Aut}(G) \), \( hnh^{-1} \) is an inner automorphism since conjugation by any automorphism maps an inner automorphism to another inner automorphism.

Key Concepts

Automorphisms

An automorphism is essentially a self-portrait of a group, painted using its own elements. Imagine a group, say, as a collection of objects with a special way to combine them. An automorphism is a complex mirror—it’s a function that reflects these objects back onto themselves in a manner respecting the group’s rules. Let's break this down:

• It's an isomorphism from a group to itself. Isomorphism means the function is bijective and respects the group operation.
• Bijective means it's both injective (one-to-one) and surjective (onto), ensuring a perfect pairing between elements of the group.
• Respecting the group operation means if you mix two objects before mirroring (applying the function), you'll get the same result as if you mixed the reflected objects after.

An example of automorphisms at work is when they collectively form a group, known as the automorphism group, under the operation of function composition.

Inner Automorphisms

Inner automorphisms are like special moves within the group—transformative but from the inside out. They are defined using an equation:For any element of the group \( x \), an inner automorphism \( \phi_g \) generated by \( g \) in the group is defined as\[\phi_g(x) = gxg^{-1}\]Here’s a closer look:

• Choose any element \( g \) in the group, multiply it by \( x \), and then multiply that product by the inverse of \( g \) (denoted \( g^{-1} \)).
• This process shuffles the group elements around while preserving their structure and identity.
• Every group element can generate its unique inner automorphism.
• All such maps from the group form the set called \( \text{Inn}(G) \).

Inner automorphisms are not just individual champs—they themselves form a subgroup within the automorphism group, binding their identities with a set purpose.

Normal Subgroup

In group theory, a normal subgroup is like a team that remains stable, even when thrown into the whirl of an operation with a larger group. If \( \text{Inn}(G) \) is a set of inner automorphisms that forms a subgroup, then it is normal if:

• For every group automorphism \( h \), \( h \text{Inn}(G) h^{-1} \) results in another set of inner automorphisms.
• This stability remains intact as their elements can smoothly move across or within these group boundaries, keeping their subgroup’s identity intact.

Conceptually, a normal subgroup can be thought of as remaining 'indifferent' to any 'conjugation' operations applied by members from outside. This concept is vital in constructing and understanding more complex structures in group theory.

Function Composition

Function composition in the realm of group theory works like seamless machinery—one function’s output is input to another, creating a chain reaction that preserves structure and rules. Specifically:

• Given two automorphisms, \( f \) and \( g \), their composition \( f \circ g \) means applying \( g \) first, then \( f \).
• Function composition creates a new function, which respects the defining rules of the original group.
• The associative property is crucial here: no matter how you pair the functions in the composition, the end result will be the same.

Automorphisms leverage function composition to form new automorphisms effortlessly, allowing them to group together to create --- literally and mathematically --- a group of transformations.