Problem 7 - Algebra: Pure and Applied

Question

Let \(G\) be a group, \(H \triangleleft G, \phi \in \operatorname{Aut}(G)\). Show that \(\phi(H) \triangleleft G\).

Step-by-Step Solution

Answer

\(\phi(H) \triangleleft G\) since \(\phi\) is an automorphism and \(H\) is normal in \(G\).

Step 1: Understand Normal Subgroups

A subgroup \(H\) of a group \(G\) is normal, denoted \(H \triangleleft G\), if for all elements \(g \in G\) and \(h \in H\), the element \(g h g^{-1} \in H\). This property needs to be preserved under the automorphism \(\phi\).

Step 2: Apply the Automorphism

An automorphism \(\phi\) is a bijective homomorphism from \(G\) to itself. This means \(\phi(gh) = \phi(g)\phi(h)\) and \(\phi(g^{-1}) = \phi(g)^{-1}\) for all \(g, h \in G\). We will use these properties to show that \(\phi(H)\) is normal in \(G\).

Step 3: Show \(\phi(H)\) is a Subgroup of \(G\)

Since \(\phi\) is an automorphism, it maps \(H\) to another subgroup \(\phi(H)\) of \(G\). As \(H\) is a subgroup, and automorphisms preserve group operations, \(\phi(H)\) retains subgroup properties.

Step 4: Demonstrate Normality Condition

For \(\phi(H)\) to be normal in \(G\), for any \(g \in G\) and \(h' \in \phi(H)\), we must have \(g h' g^{-1} \in \phi(H)\). Since \(h' = \phi(h)\) for some \(h \in H\), consider \(g \phi(h) g^{-1} = \phi(\phi^{-1}(g) h \phi^{-1}(g)^{-1})\). Since \(H\) is normal in \(G\), \(\phi^{-1}(g) h \phi^{-1}(g)^{-1} \in H\), and thus \(g \phi(h) g^{-1} \in \phi(H)\).

Step 5: Conclude That \(\phi(H) \triangleleft G\)

Given that \(g h' g^{-1} \in \phi(H)\) for any \(g \in G\) and \(h' \in \phi(H)\), the condition for normality is satisfied. Therefore, we can conclude that \(\phi(H)\) is normal in \(G\), or \(\phi(H) \triangleleft G\).

Key Concepts

Group Theory

In mathematics, group theory plays a vital role in understanding the algebraic structures known as groups. A **group** is a set combined with an operation that satisfies certain conditions—closure, associativity, identity, and invertibility. These properties ensure that the operation within the set behaves predictably.

Here's what each property means:

• Closure: If you take any two elements in your set and apply the operation, the result is still within your set.
• Associativity: When you have three elements, the order in which you apply the operation doesn't matter.
• Identity element: There exists an element in the set such that when it's combined with any element, it does not change the result.
• Inverse element: For every element in the set, there exists another element that when combined with the first under the operation results in the identity element.

Groups can vary greatly in their structure and behavior, allowing mathematicians to explore numerous possibilities within algebraic systems.

Automorphism

An automorphism is essentially a transformation of a group that maps the group onto itself, preserving its structure. In other words, it's a map from a group, say, \(G\), back to itself that retains the original group's properties.

There are key characteristics of an automorphism:

• Bijective: An automorphism must be both injective (one-to-one) and surjective (onto), meaning every element in the group is matched with a unique element and vice versa.
• Homomorphism: It respects the group operation, meaning if you take any two elements \(g\) and \(h\) in the group, \(\phi(gh) = \phi(g)\phi(h)\).

In the context of group theory, automorphisms are often used to understand symmetries and the internal structure of the group, offering insights into its properties and operations.

Subgroup Properties

A subgroup is a smaller group within a larger group, preserving the group operation and maintaining the same structural integrity. If \(H\) is a subgroup of \(G\), every operation in \(H\) must obey the same rules as those in \(G\).

There are several properties that a subgroup must have:

• It contains the identity element of \(G\).
• It is closed under the group operation.
• Each element must have its inverse also in the subgroup.

Some subgroups, known as normal subgroups, have additional properties which are crucial for certain group theory analyses. A subgroup \(H\) is normal in \(G\) if for every element \(g\) in \(G\), the element \(g h g^{-1} \) also lies in \(H\). This characteristic is fundamental when discussing quotient groups and in exploring the concept of group homomorphisms.

Bijective Homomorphism

A bijective homomorphism is a special kind of homomorphism that is both injective and surjective, essentially forming an isomorphism between groups. When we talk about a bijective homomorphism from a group to itself, we're often referring to an automorphism.

Here’s how it breaks down:

• Injective (One-to-One): Every element of the domain is mapped to a distinct element in the codomain.
• Surjective (Onto): Every element in the codomain is mapped by some element of the domain.
• Respect the operation: It ensures that the group structure is retained through \(\phi(gh) = \phi(g)\phi(h)\).

Bijective homomorphisms are central to understanding the deeper symmetries of algebraic structures, as they reveal comprehensive mappings that maintain group properties without losing any elements or creating duplicates.