Problem 28 - Algebra: Pure and Applied

Question

Let \(G\) be a group with a unique subgroup of order \(n\) and a unique subgroup of order \(m,\) where the positive integers \(n\) and \(m\) are relatively prime. Show that \(G\) has a normal subgroup of order \(n m\).

Step-by-Step Solution

Answer

The group \(G\) has a normal subgroup of order \(nm\) where \(H = H_n \times H_m\).

Step 1: Understanding the problem

We need to show that a group with unique subgroups of relatively prime orders has a normal subgroup of order equal to the product of these orders. The orders are relatively prime, meaning that their greatest common divisor is 1.

Step 2: Recall Key Definitions

A subgroup of a group is normal if it is invariant under conjugation by any element of the group. The orders being relatively prime are crucial for ensuring uniqueness.

Step 3: Use of Sylow Theorems

According to Sylow's Theorems, each Sylow subgroup is unique (and hence normal) if it is the only subgroup of its order. Since the orders are relatively prime, this implies that subgroups don't share nontrivial elements.

Step 4: Construct a Normal Subgroup with Direct Product

The uniqueness and primitivity imply both subgroups must be normal. The intersection of these subgroups is trivial. Thus, the subgroup generated by their product is isomorphic to the direct product of the two unique subgroups and has order \(n \times m\).

Step 5: Conclude with Lagrange's Theorem

By Lagrange's Theorem, since \(H_1\) and \(H_2\) are unique normal subgroups of \(G\), their orders being relatively prime means their product will be unique and hence normal in \(G\) by the product formula for finite groups.

Key Concepts

Sylow Theorems

The Sylow Theorems are named after the mathematician, Ludvig Sylow, and they are a set of theorems in group theory that provide detailed information about the number of subgroups of certain orders within a group. Specifically, the theorems focus on subgroups whose order is a power of a prime. Here's what the Sylow Theorems tell us:

• If a group has a prime power order, the existence of Sylow subgroups of that order is guaranteed.
• Every Sylow subgroup is conjugate to every other Sylow subgroup of the same order, ensuring unique behavior under group operations.
• If a Sylow subgroup is the only one of its order, then it is normal within the group.

This is crucial for solving problems involving normal subgroups, as in the original exercise, since a unique Sylow subgroup must be normal.

Normal subgroups

Normal subgroups are an integral concept in group theory, because they retain a special structure under conjugation by any element of the group. A subgroup \( H \) is normal in \( G \) if for every element \( g \) in \( G, gHg^{-1} = H \). This invariant property underlies much of group theory, as normal subgroups allow for the construction of quotient groups, which simplify the study of the larger group.In our exercise context, the group \( G \) contains unique subgroups of orders \( n \) and \( m \) respectively. Since their orders are relatively prime, they do not share nontrivial commonality with other elements of \( G \), making them normal subgroups when their order structure aligns uniquely within the group.

Relatively Prime

Two integers \( n \) and \( m \) are said to be relatively prime if their greatest common divisor (GCD) is 1. This indicates that they do not share any prime factors. This property is extremely important in number theory and group theory because it often enables independent behaviors of structures that are built from these integers.

• In groups, the property of being relatively prime ensures the non-overlapping nature of subgroup structures, leading to uniqueness.
• In the problem, the relative primeness of \( n \) and \( m \) ensures that their respective subgroups do not share elements other than the identity, leading to a product structure that is straightforward and retains normalcy in the group.

Direct Product

The concept of a direct product in group theory refers to combining two groups into a new, larger group. Assume we have two groups, \( G_1 \) and \( G_2 \), the direct product \( G_1 \times G_2 \) consists of ordered pairs \((g_1, g_2)\) where \( g_1 \in G_1 \) and \( g_2 \in G_2 \).The direct product of two groups retains the group structure with these properties:

• Each component group functions independently within the direct product.
• The order of the direct product is the product of the orders of the component groups.

In the context of the exercise, the direct product is used to construct a subgroup whose order is the product \( n \times m \), maintaining its order as normal due to unique factorization provided by relatively prime orders.

Lagrange's Theorem

Lagrange's Theorem is a fundamental theorem in group theory named after Joseph-Louis Lagrange. It states that the order (the number of elements) of any subgroup \( H \) of a finite group \( G \) divides the order of \( G \).This theorem also implies:

• The order of any group element divides the order of the group.
• If a subgroup and its order are known, you can understand the divisors of the larger group's structure.

In the exercise, Lagrange's Theorem supports the conclusion that the direct product group, formed from the relatively prime subgroups of orders \( n \) and \( m \), will have an order \( n \times m \) and be normal within \( G \), due to the uniqueness constraint and the fact their intersection is only the identity element.