Problem 4

Question

Let $G$ denote a group, and $H$ a subgroup of $G$. Prove the following: In a group $G$, a commutator is any product of the form $a b a^{-1} b^{-1}$, where $a$ and $b$ are any elements of $G .$ If a subgroup $H$ of $G$ contains all the commutators of $G$, then $H$ is normal.

Step-by-Step Solution

Verified

Answer

If a subgroup $H$ contains all the commutators of $G$, then $H$ is normal by showing $ghg^{-1} \in H$ for any $g \in G$ and $h \in H$.

1Step 1: Understand the Problem

We want to show that a subgroup $H$ of a group $G$ is normal if it contains all the commutators from $G$. A subgroup $H$ is normal in $G$ if for every $g \in G$ and $h \in H$, the element $ghg^{-1}$ is also in $H$.

2Step 2: Utilize Commutators

Recognize that by assumption, if $H$ contains all commutators of $G$, then $a b a^{-1} b^{-1} \in H$ for any $a, b \in G$. This property will help us demonstrate normality.

3Step 3: Prove Normality

To show $H$ is normal, consider any $h \in H$ and $g \in G$. We need $ghg^{-1} \in H$. Consider the element $g^{-1}hg$. By assumption, the product $(g^{-1}hg)(h^{-1})$ is a commutator when written as $x y x^{-1} y^{-1}$ and hence $gg^{-1}hg g^{-1} h^{-1} = 1 \equiv g^{-1}hg h^{-1}$ is in $H$. Thus, $ghg^{-1} = (g)(g^{-1}hg)(g^{-1}) \in H$ because $H$ is closed under products.

Key Concepts

Normal SubgroupCommutatorsSubgroupGroup

Normal Subgroup

A normal subgroup is an essential concept in Group Theory. It helps us understand the structure of groups. A subgroup $H$ of a group $G$ is called normal if it behaves nicely with respect to the group's operation. Specifically, we say $H$ is normal, denoted $H \triangleleft G$, if the following condition holds:

For every element $g$ in $G$ and every element $h$ in $H$, the element $ghg^{-1}$ is also in $H$.

This means that applying a group element $g$ to any element of $H$ (from both sides) brings you back to $H$. It allows us to form factor groups or quotient groups, which are smaller and simpler than the original group.

Quotient groups help analyze group structure by breaking down complicated groups into simpler parts.
Normal subgroups are crucial when working with homomorphisms, another fundamental topic in Group Theory.

Commutators

Commutators are special elements in a group that help us measure how close the group is to being abelian, meaning commutative. The commutator of two elements $a$ and $b$ in a group $G$ is defined as:

$[a, b] = a b a^{-1} b^{-1}$

If $G$ is an abelian group, the commutator $[a, b]$ will always equal the identity element. When a subgroup $H$ contains all the commutators of $G$, it represents a good measure of this non-commutativity. In fact, the subgroup generated by all commutators is known as the derived subgroup or the commutator subgroup of $G$.

This subgroup is crucial in determining how a group can be simplified or understood in terms of its abelian properties.
For understanding Normality: If $H$ contains all commutators, it helps prove that $H$ must be normal.

Subgroup

A subgroup is simply a smaller group that lives inside a larger group. Formally, if $H$ is a subgroup of a group $G$, then $H$ must satisfy three key properties:

Closure: For any $a, b \in H$, the product $ab$ must also be in $H$.
Identity: The identity element $e$ of $G$ must be in $H$.
Inverses: For every $a \in H$, its inverse $a^{-1}$ must also be in $H$.

These properties ensure that $H$ contains everything necessary to still operate fully as a group but within the restrictions of $G$. Subgroups are essential building blocks in Group Theory.

They allow us to create and analyze structural pieces within larger groups.
Understanding subgroups lets us explore normal subgroups and hence quotient groups.

Group

The concept of a group is fundamental in mathematics, specifically in algebra. A group is a mathematical structure used to analyze symmetry, operations, and transformations. A group $G$ consists of a set of elements and an operation that satisfies four central properties:

Closure: Performing the operation on any two elements in $G$ results in another element in $G$.
Associativity: The group operation is associative; that is, for any $a, b, c \in G$, the equation $(ab)c = a(bc)$ holds.
Identity: There exists an identity element $e$ in $G$ such that for any element $a \in G$, the equation $ae = ea = a$ holds.
Inverses: For each element $a$ in $G$, there exists an element $b$ in $G$ such that $ab = ba = e$, where $e$ is the identity element.

Groups are ubiquitous in mathematics due to their utility in simplifying and understanding mathematical structures.

Groups provide insight into the symmetries and transformations possible within a mathematical object or system.
They are foundational to many branches of mathematics, including algebra, geometry, and number theory.

Problem 4

Other exercises in this chapter

Problem 4

A property of groups is said to be "preserved under homomorphism" if, whenever a group $G$ has that property, every homomorphic image of $G$ does also. In t

View solution

Problem 4

If $f: G \rightarrow H$ is a homomorphism, prove each of the following: Let $m$ be an integer such that $m$ and $|H|$ are relatively prime. For any \(x

View solution

Problem 4

Let $G, H$, and $K$ be groups. Prove the following: If $f: G \rightarrow H$ is a homomorphism and $J$ is any subgroup of $H$, then- $$ f^{-1}(J)=\\{x

View solution

Problem 4

Imagine a square as a piece of paper lying on a table. The side facing you is side A. The side hidden from view is side $B$. Every motion of the square either

View solution