Question

Sei \(f: G \rightarrow H\) ein Gruppenhomomorphismus. Zeigen Sie, dass für jedes Element \(g\) von \(G\) gilt: \(f\left(g^{-1}\right)=f(g)^{-1}\).

Short answer

The reverse image of the inverse under a group homomorphism is the inverse of the image: \( f(g^{-1}) = f(g)^{-1} \).

Step-by-step solution

Understand the Definition of a Group Homomorphism

A group homomorphism is a function \( f: G \rightarrow H \) between two groups \( G \) and \( H \) such that for all elements \( a, b \in G \), it holds that \( f(ab) = f(a)f(b) \). This property is crucial to the solution.

Apply Group Homomorphism Definition to Inverse

To prove the property \( f(g^{-1}) = f(g)^{-1} \), we start by recognizing that \( gg^{-1} = e_G \), where \( e_G \) is the identity element in the group \( G \).

Homomorphism Applied to Group Identity

Using the homomorphism property, we apply \( f \) to both sides of the equation \( gg^{-1} = e_G \). This gives us \( f(gg^{-1}) = f(e_G) \).

Simplify Using Homomorphism Property

The left-hand side can be simplified using the homomorphism property: \( f(gg^{-1}) = f(g)f(g^{-1}) \). Thus, we have \( f(g)f(g^{-1}) = f(e_G) \).

Identity Equation in Target Group

In any group \( H \), the identity element, \( e_H \), satisfies \( f(e_G) = e_H \). Hence, our equation becomes \( f(g)f(g^{-1}) = e_H \).

Solve for \( f(g^{-1}) \)

From the equation \( f(g)f(g^{-1}) = e_H \), isolate \( f(g^{-1}) \) by multiplying both sides on the left by \( f(g)^{-1} \), which is the inverse of \( f(g) \) in \( H \): \( f(g)^{-1}f(g)f(g^{-1}) = f(g)^{-1}e_H \).

Conclude with Identity Properties

Since \( f(g)^{-1}f(g) = e_H \), the left side simplifies to \( f(g^{-1}) = f(g)^{-1} \). This completes the proof that \( f(g^{-1}) = f(g)^{-1} \) for any \( g \in G \).

Key concepts

Inverse Elements

In group theory, the concept of an "inverse element" is so crucial that it can determine the structure and characteristics of a group. To grasp this idea, it's important to understand what an inverse element does.

• In a group, every element must have an inverse element.
• This means for any element \( g \) in the group, there exists another element \( g^{-1} \) such that when they are combined, they yield the identity element: \( gg^{-1} = e \) and \( g^{-1}g = e \).
• It's like having a pair that cancels each other out to bring us back to the starting point, or the identity element.

When dealing with group homomorphisms, the inverse relation is preserved between the groups. A great example is when applying a homomorphism \( f: G \rightarrow H \), if \( g^{-1} \) is the inverse of \( g \) in group \( G \), then \( f(g^{-1}) \) should be equivalent to \( f(g)^{-1} \), the inverse in group \( H \). This is a key property that maintains the group structure across different groups through homomorphisms.

Identity Element

The identity element is a fundamental building block in group theory. It's essentially the 'do nothing' or 'neutral' element that doesn't change other elements when combined with them.

• In any group, for every element \( a \), the equation \( ae = ea = a \) holds true, where \( e \) is the identity element of the group.
• It acts as a central anchor, ensuring that when you multiply or combine an element by the identity, the element remains unchanged.
• Think of it as zero in addition or one in multiplication; it keeps everything steady.

When considering a group homomorphism, this identity element plays a crucial role. If you apply a homomorphism \( f: G \rightarrow H \) to the identity element of group \( G \), the result is the identity element of group \( H \). This property ensures the homomorphism respects and preserves this fundamental aspect of group structure.

Group Theory

Group theory is a cornerstone of abstract algebra, providing a framework to study symmetry, structure, and transformations. At its heart, a group is composed of elements combined under a binary operation that respects four main principles:

• Closure: For any elements \( a, b \) in the group, the result of the operation \( a \, * \, b \) is also in the group.
• Associativity: The group operation is associative, meaning \( (a \, * \, b) \, * \, c = a \, * \, (b \, * \, c) \).
• Identity Element: The group must have an identity element that, when combined with any element \( g \), results in \( g \).
• Inverse Elements: Each element \( g \) must have an inverse \( g^{-1} \) such that \( g \, * \, g^{-1} = e \).

Understanding the interplay of these properties is essential to mastering group theory. Group homomorphisms, like the one given from group \( G \) to group \( H \), are functions that maintain these structural properties between groups, preserving the relationships like identities and inverses. This makes group theory a powerful tool to analyze mathematical structures and their transformations across different domains.