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 \(\phi(H)=H\). Show that if \(H\) is a characteristic subgroup of \(G,\) then \(H \triangleleft G\).

Short answer

If \( H \) is characteristic in \( G \), then \( H \) is normal in \( G \) since every inner automorphism, being a special automorphism, maps \( H \) to itself.

Step-by-step solution

Understanding the Definition

To begin, we need to recognize that a characteristic subgroup \( H \) of a group \( G \) means that for every automorphism \( \phi \) in \( \text{Aut}(G) \), we have \( \phi(H) = H \). An automorphism is a bijective homomorphism from the group to itself. Our goal is to show that \( H \) is also a normal subgroup of \( G \), which means that for all elements \( g \) in \( G \), \( gHg^{-1} = H \).

Show Automorphism Implication

Consider any inner automorphism \( \phi_g \) defined by \( \phi_g(x) = gxg^{-1} \) for all \( x \in G \). This is a special type of automorphism where the function maps \( x \) to its conjugate by \( g \). If \( H \) is characteristic, then \( \phi_g(H) = H \).

Use Characteristic Property

Since \( H \) is characteristic in \( G \), it means specifically that \( \phi_g(H) = H \) for the inner automorphism \( \phi_g \). Therefore, for each \( g \in G \), applying the inner automorphism \( \phi_g \) results in \( gHg^{-1} = H \). This satisfies the condition for \( H \) being a normal subgroup of \( G \).

Conclude Normal Subgroup Status

Because \( gHg^{-1} = H \) for every \( g \in G \), \( H \) must be a normal subgroup of \( G \) by definition. Thus, we have shown that if \( H \) is a characteristic subgroup of \( G \), then necessarily \( H \triangleleft G \).

Key concepts

Normal Subgroup

In group theory, a normal subgroup is a subgroup that stays consistent across the entire group through conjugation.
What this means is, if you take any element from the group and use it to conjugate an element from the subgroup, it will result in another element that still belongs to the subgroup.
In formal terms, a subgroup \( H \) of a group \( G \) is normal if for every element \( g \) in \( G \), the equation \( gHg^{-1} = H \) holds true.
This property is crucial because normal subgroups are the foundation for constructing quotient groups, which are essential in simplifying complex group structures.

• Normal subgroups are invariant under conjugation by any group element.
• They help in defining quotient groups.
• Every characteristic subgroup is a normal subgroup, but not all normal subgroups are characteristic.

Automorphism

An automorphism is a mapping of a group to itself that keeps the group's structure intact.
It's a bijective homomorphism, meaning it is both one-to-one and onto, ensuring that each element and its image correspond to an element of the other set.
Automorphisms are critical in understanding the symmetries within a group as they reveal how one part of the group can mirror another while preserving all the group's operational aspects.
For a group \( G \), an automorphism \( \phi \) would transform the group such that for all elements \( x, y \) in \( G \), \( \phi(xy) = \phi(x)\phi(y) \).

• Automorphisms can exhibit many forms of symmetry within a group.
• The set of all automorphisms of a group forms a group itself, known as the automorphism group.
• They provide insights into the group’s structure and its possible transformations.

Inner Automorphism

Inner automorphisms serve as a specific type of automorphism, where each element of a group defines its own unique transformation.
For any element \( g \) in a group \( G \), an inner automorphism is determined by the formula \( \phi_g(x) = gxg^{-1} \) for any \( x \) in \( G \).
This transformation essentially "shuffles" the group elements using conjugation, showing one way the group can "reshape" itself while still remaining the same in structure.
Understanding inner automorphisms is key to grasping how groups internally rearrange while maintaining their inherent properties like closure and associativity.

• Inner automorphisms reflect the group's internal structure and symmetry.
• They are determined by conjugation by elements of the group.
• Every inner automorphism is an automorphism, but not vice versa.

Group Theory

Group theory studies the algebraic structures known as groups, which consist of a set of elements paired with an operation that satisfies certain conditions such as closure, associativity, identity, and invertibility.
This concept forms the backbone of abstract algebra, providing the framework to explore symmetries and transformations within mathematical systems and beyond.
Groups themselves can range from simple sets, like integers under addition, to highly complex structures used in advanced mathematical fields.
Understanding group theory is vital for grasping everything from crystallography to advanced physics.

• Basic properties every group must satisfy are closure, associativity, identity, and invertibility.
• Group theory aids in analyzing and constructing mathematical structures.
• It has vast applications across mathematics, sciences, and engineering.