Question

A subgroup \(H\) of a group \(G\) is called a characteristic subgroup of \(G\) if for all \(\phi \in \operatorname{Aut}(G)\) we have \(\phi(H)=H\). Let \(G\) be a group, \(H\) a normal subgroup of \(G,\) and \(K\) a characteristic subgroup of \(H .\) Show that \(K\) is a normal subgroup of \(G\).

Short answer

Since K is characteristic in H and H is normal in G, K is normal in G.

Step-by-step solution

Define normal and characteristic subgroups

A normal subgroup \(H\) of a group \(G\) is a subgroup such that for all \(g \in G\), \(gHg^{-1} = H\). A characteristic subgroup \(K\) of a subgroup \(H\) is a subgroup such that for every automorphism \(\phi \in \text{Aut}(H)\), \(\phi(K) = K\). Our goal is to show that \(K\) is normal in \(G\).

Use the normality of H in G to relate elements

Since \(H\) is a normal subgroup of \(G\), for any \(g \in G\) and \(h \in H\), we have \(ghg^{-1} \in H\). This implies \(gHg^{-1} = H\), and hence any conjugation by an element \(g\in G\) induces an automorphism of \(H\).

Relation between characteristic property and normality

Since \(K\) is characteristic in \(H\), for any automorphism \(\psi \in \text{Aut}(H)\), specifically the one induced by \(g \in G\) (from Step 2), we have \(\psi(K) = K\). Therefore, \(gKg^{-1} \subset K\). This must hold for all \(g \in G\).

Conclusion

Since \(gKg^{-1} = K\) for any element \(g \in G\), \(K\) satisfies the normality condition \(gKg^{-1} = K\). Therefore, \(K\) is a normal subgroup of \(G\).

Key concepts

Normal Subgroup

In group theory, a normal subgroup is quite an important concept. It helps us understand the structure and symmetry of a mathematical group. When a subgroup \( H \) of a group \( G \) is normal, it means that it interacts nicely with the elements of \( G \). More specifically, for every element \( g \) in \( G \), when you conjugate \( H \) by \( g \) (which means measuring \( gHg^{-1} \)), it always results in \( H \) itself. This means the subgroup \( H \) looks the same even after this transformation.

Normal subgroups play a critical role in the construction of quotient groups, which are fundamental in understanding the concept of simple groups and how a group can be broken down into smaller components. Understanding the behavior of normal subgroups helps us classify and analyze groups deeply, as it guarantees certain symmetries and properties to hold.

Here's a quick summary of key features of a normal subgroup:

• Invariant under conjugation by elements of the parent group.
• Essential for forming quotient groups.
• Every subgroup of an abelian group is normal, as \( gHg^{-1} = H \).

Group Theory

Group theory is a branch of mathematics that deals with algebraic structures known as groups. At its core, group theory studies the symmetry and structure of mathematical objects through the lens of a set equipped with an operation that satisfies certain axioms: closure, associativity, identity, and inversibility.

To give you an idea of the importance of group theory, think of it as a powerful tool allowing mathematicians and scientists to explore and organize the symmetries they encounter across different fields. Whether it's solving a Rubik's Cube, analyzing the structure of molecules, or understanding fundamental physics, group theory has far-reaching implications.

Some fundamental aspects of group theory include:

• Closure: If you take two elements from a group and combine them through the group operation, the result stays within the group.
• Associativity: The way you group elements during an operation doesn’t change the result.
• Identity element: There is an element in every group which, when combined with any element, leaves the other element unchanged.
• Inverse element: For each element in a group, there's another element that can "undo" its effect through the group operation.

Group theory helps us uncover the essence of symmetry in mathematical contexts and lets us apply these uniform principles across diverse domains.

Automorphism of a Group

An automorphism is a fascinating concept in group theory, describing a special kind of symmetry within the group itself. When we apply an automorphism to a group, we are essentially finding a way to rearrange the group such that its structure remains unchanged.

Think of an automorphism as a special function or mapping \( f: G \to G \) that takes elements from the group \( G \) and shifts them around, while ensuring that the group's operation still behaves in exactly the same manner. This function is bijective, meaning it’s both one-to-one and onto. Thus, each distinct element in \( G \) maps to another distinct element, and every possible element in \( G \) is covered.

Understanding automorphisms is crucial as they help identify whether two groups are similar in structure. These mappings show that despite different appearances, they might be fundamentally the same. Key features to remember about automorphisms are:

• They preserve group operation—if \( a \cdot b = c \), then \( f(a) \cdot f(b) = f(c) \).
• They maintain the identity and inverse elements unchanged, meaning \( f(e) = e \) and \( f(a^{-1}) = (f(a))^{-1} \).
• The set of all automorphisms of a group \( G \), denoted by \( \text{Aut}(G) \), itself forms a group under composition, revealing even more about the group’s internal symmetry.

Recognizing and exploring automorphisms expands our ability to examine the intrinsic qualities of mathematical groups.