Problem 34 - A First Course in Abstract Algebra

Question

Show that if a finite group \(G\) has exactly one subgroup \(H\) of a given order, then \(H\) is a normal subgroup of \(G\).

Step-by-Step Solution

Answer

The subgroup \(H\) is normal in \(G\) since \(gHg^{-1} = H\) for all \(g \in G\).

Step 1: Understand the Problem Statement

We need to prove that if a finite group \(G\) has exactly one subgroup \(H\) of a given order, then \(H\) is a normal subgroup of \(G\). This means that \(gHg^{-1} = H\) for all elements \(g\) in \(G\).

Step 2: Recall the Definition of a Normal Subgroup

A subgroup \(H\) of a group \(G\) is called a normal subgroup if for all elements \(g\) in \(G\), conjugation by \(g\) leaves \(H\) invariant. Mathematically, this is expressed as \(gHg^{-1} = H\).

Step 3: Use the Property of Conjugate Subgroups

Recall that for any subgroup \(H\) and element \(g \in G\), the set \(gHg^{-1}\) is also a subgroup of \(G\) that has the same order as \(H\).

Step 4: Analyze the Given Condition

Since \(G\) is stated to have exactly one subgroup \(H\) of the given order, and \(gHg^{-1}\) is a subgroup of \(G\) with the same order as \(H\), it follows that \(gHg^{-1}\) must be \(H\) itself. This is because there cannot be another subgroup of the same order.

Step 5: Conclude that \(H\) is Normal

Given that \(gHg^{-1} = H\) for all \(g \in G\), by definition, \(H\) is normal in \(G\). Therefore, whenever a finite group has a unique subgroup of a certain order, that subgroup is normal in the group.

Key Concepts

Finite Groups

In mathematics, a finite group is a set equipped with an operation that combines any two of its elements to form another element, and that set has a finite number of elements. Such groups satisfy four main properties: closure, associativity, an identity element, and invertibility.
These properties together form the algebraic structure known as a group. When we deal with finite groups, the number of elements, or "order," plays a significant role in determining the group's structure and behavior.
The concept of a finite group is crucial in various areas of mathematics and applied sciences because it simplifies many algebraic structures by narrowing the realm of possibilities.

Normal Subgroup

A normal subgroup is a special type of subgroup within a group that remains invariant under conjugation by any element of the group. When you have a subgroup \( H \) of a larger group \( G \), \( H \) is a normal subgroup if for every element \( g \) in \( G \), the set resulting from \( gHg^{-1} \) is exactly \( H \) again.
In symbolic terms, this relationship can be expressed as \( gHg^{-1} = H \).
Why does normality matter? Normal subgroups are fundamental in the study of group theory because they allow us to construct other groups, called quotient groups, from a given group.
These quotient groups reveal a lot about the structure of the original group \( G \), helping to understand its properties and classification.

Conjugate Subgroups

When you talk about conjugate subgroups, you're referring to subgroups that have the same internal structure but might differ based on their position within a group. If you take a subgroup \( H \) in a group \( G \) and apply a transformation with an element \( g \) in \( G \), forming \( gHg^{-1} \), you get another subgroup within \( G \), which is a conjugate of \( H \).
Each conjugate of \( H \) has the same order and properties as \( H \) itself but might be physically shifted within the group \( G \).
Understanding conjugate subgroups is integral to recognizing symmetrical properties and invariant structures within groups.

• Conjugates indicate the ways a subgroup can relate to itself when transformed within the group's structure.
• Studying these structures helps in understanding how groups behave under various operations, which is essential for more advanced group theories, like representation theory.

Subgroup Order

The order of a subgroup refers to the number of elements it contains. This property is fundamental because in group theory, the order of subgroups and their relation to the group's order itself (Lagrange's theorem) provides insights into the compositions and possibilities within a group.
If a finite group \( G \) has, say, 12 elements, any subgroup of \( G \) must have an order that divides 12. This logical deduction stems from Lagrange's theorem, which states that the order of any subgroup must be a divisor of the order of the whole group.
Understanding subgroup order helps in determining potential normal subgroups or figuring out unique properties and solutions related to group exercises.

• In situations where a subgroup of a particular order is unique, that uniqueness often implies additional properties, such as normality within the larger group.
• Analyzing implications of uniqueness in subgroup order can unveil deeper relationships and constraints in group structure.