Problem 5 - A Book of Abstract Algebra

Question

In the following, let \(G\) denote an arbitrary group. Let \(H\) be a subgroup of \(G . H\) is normal iff \(a H=H a\) for every \(a \in G\).

Step-by-Step Solution

Answer

A subgroup \(H\) is normal in \(G\) if and only if \(aH = Ha\) for every \(a \in G\).

Step 1: Understanding Normal Subgroups

In group theory, a subgroup \(H\) is normal in a group \(G\) if and only if \(aH = Ha\) for all elements \(a\) in \(G\). This means that \(H\) is invariant under conjugation by any element from \(G\). Essentially, \(aH = \{ah | h \in H\}\) should equal \(Ha = \{ha | h \in H\}\).

Step 2: Condition of Normal Subgroup

To prove \(H\) is normal if and only if \(aH = Ha\), assume first that \(H\) is normal in \(G\). By definition of normality, \(aH = Ha\) follows directly for every \(a \in G\). Therefore, if \(H\) is normal, then conjugation \(aHa^{-1} = H\) holds, proving \(aH = Ha\).

Step 3: Converse Part of the Proof

For the converse, assume \(aH = Ha\) for all \(a\in G\). Then for any element \(a\) and any \(h \in H\), \(aha^{-1} \in H\) because \(ah \in aH = Ha\), meaning there exists some \(h' \in H\) such that \(ah = h'a\). Therefore, \(aha^{-1} = h' \in H\), and \(H\) is normal in \(G\).

Step 4: Conclusion

The condition \(aH = Ha\) for all elements \(a\in G\) is both necessary and sufficient for \(H\) to be normal in \(G\). This completes the proof.

Key Concepts

Group Theory

Group Theory is a branch of mathematics that studies sets equipped with an operation that satisfies certain axioms. These include closure, associativity, identity, and invertibility. A group consists of a set and an operation that combines any two elements of the set to form another element of the same set. Groups provide a way to abstract and capture the essence of symmetry.

• Closure: If you take any two elements in the group and apply the group operation, the result is also in the group.
• Associativity: The way in which the elements are grouped in executing the operation does not affect the outcome.
• Identity Element: There exists an element in the group such that any element in the group combined with the identity element return the element itself.
• Inverse Element: For each element in the group, there exists another element such that their product is the identity element.

This foundational framework supports many areas of mathematics and its applications. By understanding the structure and properties of groups, one can gain insight into the transformations and symmetries within different mathematical and physical systems.
An important concept in group theory is that of normal subgroups, which we will explore further.

Subgroup

In group theory, a subgroup is a subset of a group that is itself a group under the same operation. Let's break this down simply: If you have a group, named, say, \( G \), a subset of this group, \( H \), can be called a subgroup if:

• \( H \) is non-empty.
• \( H \) is closed under the group operation. If you take any two elements from \( H \) and perform the group operation, you'll still be within \( H \).
• \( H \) contains the identity element of \( G \).
• \( H \) includes the inverse element for each of its elements.

Not every subset of a group is automatically a subgroup. These conditions ensure that a subgroup maintains the group structure and respects the properties shared by the whole group.
Subgroups are like smaller versions of the group, sharing the same important features.

Conjugation

Conjugation in group theory refers to a specific method of combining an element from the group with another from its subgroup. If we have two elements, \( a \) and \( h \), from the group \( G \), conjugation typically looks like this: \( aha^{-1} \).
Conjugation can be thought of as applying "a transformation" to the element \( h \) via element \( a \) and then reversing that transformation. This process is central to many aspects of group theory:

• Symmetry and Structure: Conjugation can reveal symmetries and structures within groups, providing insight into their behavior.
• Equivalence: Two elements are conjugate if one is a transformation of the other. They share many group-theoretic properties.

Conjugation helps us understand how elements in the group act on each other and can be key in identifying normal subgroups, where every conjugate of an element from the subgroup remains within the subgroup.

Invariant under conjugation

When we say a subgroup is invariant under conjugation, it means that if you take any element from the subgroup and perform the conjugate operation with any element from the entire group, the result still lies within the subgroup.

• This property defines a normal subgroup. If \( H \) is a normal subgroup of \( G \), then for every element \( a \) in \( G \) and every element \( h \) in \( H \), the element \( aha^{-1} \) will also be in \( H \).
• This feature is vital because it ensures a certain kind of stability and symmetry; rearranging parts of the subgroup through conjugation doesn't take you outside of it.

Understanding invariance under conjugation helps to not only recognize normal subgroups but also appreciate their importance in the wider framework of group theory.
These subgroups provide a bridge to more complex structures and deeper insights in mathematical systems.