Problem 17 - A First Course in Abstract Algebra

Question

Show that every group of onder \((35)^{3}\) has a normal subgroup of order \(125 .\)

Step-by-Step Solution

Answer

The group has a normal subgroup of order 125 due to the uniqueness of the Sylow 5-subgroup.

Step 1: Understand the Problem

The problem requires showing that a group with order \((35)^3\) has a normal subgroup of order 125. This involves using Sylow theorems and properties of groups.

Step 2: Calculate the Order of the Group

First, calculate the order of the group: \( (35)^3 = (5 imes 7)^3 = 35^3 = 5^3 imes 7^3 = 125 imes 343 \). So the group has order 42,875.

Step 3: Apply Sylow's Theorems for Subgroup Existence

According to Sylow's Theorems, for each prime factor, there exists a subgroup whose order is a power of that prime. Here, the prime factors are 5 and 7.

Step 4: Calculate Subgroup Orders

For prime 5, the possible order of Sylow subgroup is \(5^3 = 125\). For prime 7, the possible order of Sylow subgroup is \(7^3 = 343\).

Step 5: Determine the Number of Sylow Subgroups

The number of Sylow \(p\)-subgroups, denoted \(n_p\), satisfies specific congruence properties (modulo the prime), and divides the group order. Specifically, \(n_5 \equiv 1 \pmod{5}\) and \(n_5\) divides 343.

Step 6: Check for Normal Subgroup Existence

\(n_5\) can only be 1 or a multiple ensuring \(n_5\equiv 1 \pmod{5}\). Since \(n_5 = 1\) divides 343 and satisfies the congruence, there is essentially 1 Sylow \(5\)-subgroup, making it normal.

Key Concepts

Sylow theorems

Sylow theorems are a fundamental set of theorems in group theory that help us understand the composition of groups by analyzing the existence and number of subgroups of particular orders. These theorems are applicable exclusively to finite groups and offer insights into the possible configurations of subgroups.

They let us determine:

• For a given prime number \( p \), if there's a subgroup whose order is a power of \( p \).
• The number of such subgroups, which must meet two criteria: it divides the group order and the formula \( n_p \equiv 1 \pmod{p} \) holds true.

Thus, if a group order is expressed in terms of its prime factors, Sylow theorems can be instrumental in dissecting that group into smaller, more manageable subgroups. They play a key role in proving the existence of normal subgroups under specific conditions, as shown in the exercise.

normal subgroup

A normal subgroup is a special type of subgroup whose structure enables seamless interactions within a larger group. In a group \( G \), a subgroup \( N \) is termed as normal if it remains invariant under the action of conjugation by any element of \( G \).

This invariance property means that for any \( g \in G \) and \( n \in N \):

• The product \( g n g^{-1} \) will also be in \( N \).

Normal subgroups are significant because they allow the extension of group operations to form quotient groups. These quotient groups help simplify group analysis and prove fundamental results like in the exercise where we deduce the presence of a normal subgroup using Sylow theorems. Detecting normal subgroups aids in characterizing group structure and understanding its symmetry properties.

prime factorization

Prime factorization breaks down a number into the product of its prime numbers, revealing its fundamental structure. In the context of groups, it allows mathematicians to scrutinize a group's order by representing it as a product of powers of primes.

For instance, in the exercise provided, the group's order \(35^3\) is factored as \( 5^3 \times 7^3 \), indicating that any subgroup must have orders corresponding to these factors. This decomposition is essential when applying results such as Sylow theorems to ascertain subgroup characteristics. The factorization unveils the fundamental building blocks of the group, guiding mathematicians in examining the specific prime power combinations possible within its subgroups.

p-groups

In group theory, a \( p \)-group is a group where each element's order (the least positive integer such that the element raised to this power equals the group's identity) is a power of a prime \( p \). These groups have properties that make them particularly manageable under certain mathematical frameworks.

Key characteristics include:

• Being abelian if the group order is \( p^2 \) or less.
• Exhibiting intensive symmetry and algebraic regularity due to their power-related order structure.

In our exercise, identifying a \( 5 \)-group, with order \( 5^3 = 125 \), helps establish a subgroup with specific characteristics as guided by Sylow's theorems. The simplicity and structured nature of \( p \)-groups make them a pivotal concept in elucidating the behavior and classification of larger abstract groups.