Problem 27 - Algebra: Pure and Applied

Question

Let \(G\) be a group and let \(S=\left\\{x y x^{1} y^{-1} \mid x, y \in G\right\\} .\) Let \(N\) be the subgroup \(\langle S\rangle\) generated by \(S\) (in the sense of Exercise 25 ), called the commutator subgroup of \(G\). Show that (a) \(N \triangleleft G\) (b) \(G / N\) is Abelian. (c) If \(H\) is a normal subgroup of \(G\) and \(G / H\) is Abelian, then \(N \subseteq H\). (d) If \(H\) is a subgroup of \(G\) with \(N \subseteq H\), then \(H \triangleleft G\).

Step-by-Step Solution

Answer

(a) N is normal in G. (b) G/N is Abelian. (c) N ⊆ H if G/H is Abelian and H is normal. (d) N ⊆ H implies H is normal in G.

Step 1: Understand Commutator Subgroups

The commutator subgroup, denoted as \(N\), of a group \(G\) is the subgroup generated by all commutators, which are elements in the form \(x y x^{-1} y^{-1}\) for \(x, y \in G\). This subgroup captures the notion of 'how far' a group is from being Abelian. Commutators measure the 'non-commutativity' of the group.

Step 2: Show N is a Normal Subgroup (a)

A subgroup \(N\) is normal in \(G\) if for every \(g \in G\) and every \(n \in N\), the element \(gng^{-1} \in N\). Since \(N\) is generated by elements of the form \(x y x^{-1} y^{-1}\), and these elements commute under conjugation (meaning \((g(x y x^{-1} y^{-1})g^{-1} = (gx) (gy) (gx)^{-1} (gy)^{-1})\)), it follows that \(N\) is normal in \(G\).

Step 3: Prove G/N is Abelian (b)

To show \(G/N\) is Abelian, consider any two elements \(gN, hN \in G/N\). Their product is \((gN)(hN) = ghN\), and since commutators are in \(N\), \((gN)(hN) = ghN = hgN = (hN)(gN)\). This implies addition in \(G/N\) is commutative, so \(G/N\) is Abelian.

Step 4: Show N ⊆ H for normal H (c)

Given \(H\) is normal in \(G\) and \(G/H\) is Abelian, all commutators in \(G\) must be in \(H\) because elements commute modulo \(H\). This means every generator of \(N\) is contained in \(H\), hence \(N\subseteq H\).

Step 5: Show H is Normal if N ⊆ H (d)

If \(N \subseteq H\) and \(H\) is a subgroup of \(G\), for \(H\) to be normal in \(G\), we must have \(ghg^{-1} \in H\) for every \(h \in H\) and \(g \in G\). Since \(N\) contains all commutators and \(G/N\) is Abelian, conjugation by elements from \(G\) induces commutation, ensuring that pieces of commutation like \(ghg^{-1}\) remain within \(N\). Since \(N \subseteq H\), this implies conjugation results \(ghg^{-1} \in H\).

Key Concepts

Commutator Subgroups

Commutator subgroups are an essential concept in group theory, designed to capture the idea of how close a group is to being Abelian. In simpler terms, a commutator subgroup contains all the 'commutator' elements from the group. A commutator is an expression of the form \(xyx^{-1}y^{-1}\), where \(x\) and \(y\) are elements of the group \(G\). The commutator subgroup, often denoted as \(N\), is generated by all such commutators.

This concept is fascinating because it measures the non-commutativity of a group. If a group is Abelian, its elements commute, meaning \(xy = yx\). Therefore, every commutator in an Abelian group equals the identity element, making the commutator subgroup trivial (consisting only of the identity element). The larger the commutator subgroup, the more non-Abelian the group is. Understanding commutator subgroups is pivotal in determining the structure of a group.

Normal Subgroups

Normal subgroups are subgroups that remain stable under conjugation by any element of the parent group. For a subgroup \(N\) to be normal in \(G\), for every element \(g\) in \(G\) and every element \(n\) in \(N\), the conjugated element \(gng^{-1}\) must also be in \(N\).

This property is extremely useful for many mathematical constructs and theorems, particularly since normal subgroups aid in defining quotient groups. Quotient groups, formed when a group is divided by a normal subgroup, play a crucial role in group homomorphisms and symmetry operations.

In the context of the commutator subgroup, we typically prove that it is a normal subgroup of \(G\). This is because commutators themselves do not change form under conjugation, yielding a structure that fits neatly into the normal subgroup definition. This property of being a "normal" subgroup makes commutator subgroups an invaluable tool in simplifying and understanding the operations within groups.

Abelian Groups

Abelian groups, named after the mathematician Niels Henrik Abel, are groups where the group operation is commutative. This means that for any two elements \(a\) and \(b\) in an Abelian group \(G\), the equation \(ab = ba\) always holds true.

Commutativity is a fascinating property since it simplifies the structure of a group significantly. Many familiar systems, like the set of integers under addition, are Abelian. When you consider quotient groups like \(G/N\) where \(N\) is the commutator subgroup of \(G\), it results in an Abelian group. This arises because the effect of non-commuting elements is absorbed into \(N\), leaving the remaining structure to commute freely.

Understanding which groups are Abelian helps in categorizing their behavior and predicting how they interact with other mathematical structures. Abelian groups have well-studied and often simpler properties that make them easier to analyze than their non-Abelian counterparts.

Group Homomorphisms

Group homomorphisms are structure-preserving maps between two groups. A homomorphism from group \(G\) to group \(H\) ensures that the group operation is preserved. Specifically, if \(f: G \to H\) is a homomorphism, then for all elements \(a, b\in G\), the map ensures \(f(ab) = f(a)f(b)\).

In the context of normal subgroups and quotient groups, group homomorphisms are incredibly powerful. They help create bridges between different groups, allowing us to understand their structure and interrelations better. In particular, if a subgroup is normal, it can be the kernel of a homomorphism, linking it directly to quotient group constructions.

By assessing group homomorphisms, mathematicians can uncover symmetry properties and leverage different groups' structures to solve complex problems. Understanding homomorphisms is foundational to advancing in group theory and its applications.