Problem 6 - A Book of Abstract Algebra

Question

Let \(G\) and \(H\) be groups, with \(a \in G\) and \(b \in H .\) Prove the following: Suppose \((c, d) \in G \times H\), where \(c\) has order \(m\) and \(d\) has order \(n .\) If \(m\) and \(n\) are not relatively prime (hence have a common factor \(q>1)\), then the order of \((c, d)\) is less than \(m n\).

Step-by-Step Solution

Answer

The order of \((c, d)\) is less than \(mn\) since \(\gcd(m, n)\) is greater than 1.

Step 1: Understand the Problem Statement

We need to prove that if two elements \(c\) and \(d\) in groups \(G\) and \(H\), respectively, have orders \(m\) and \(n\) that are not relatively prime, then the order of the element \((c, d)\) in the product group \(G \times H\) is less than \(mn\).

Step 2: Clarify the Consequence of Non-Relatively Prime Orders

Since \(m\) and \(n\) are not relatively prime, they have a common divisor \(q > 1\) such that \(q \mid m\) and \(q \mid n\). This means that \(m = kq\) and \(n = lq\) for some integers \(k\) and \(l\).

Step 3: Consider the Orders in the Product Group

In the group \(G \times H\), the order of a general element \((c, d)\) is given by the least common multiple of the orders of \(c\) and \(d\), i.e., \(\operatorname{lcm}(m, n)\).

Step 4: Calculate the LCM of \(m\) and \(n\)

Since \(q\) is a common factor of both \(m\) and \(n\), we have \(\operatorname{lcm}(m, n) = \frac{m \cdot n}{\gcd(m, n)}\). The gcd is at least \(q\), so \(\operatorname{lcm}(m, n) = \frac{m \cdot n}{q} \leq mn/q < mn\).

Step 5: Conclude the Order of \((c, d)\)

Since \(\operatorname{lcm}(m, n) < mn\), the order of \((c, d)\) is less than \(mn\). Therefore, the order of \((c, d)\) in \(G \times H\) is strictly less than the product of the orders of \(c\) and \(d\).

Step 6: Final Conclusion

We have shown that because \(m\) and \(n\) are not relatively prime, \((c, d)\) has order of \(\operatorname{lcm}(m, n)\) which is less than \(mn\), completing the proof.

Key Concepts

Order of an Element

In group theory, the **order of an element** is a fundamental concept that refers to the smallest positive integer \(k\) for which \(a^k = e\), where \(e\) is the identity element in the group and \(a\) is an element of the group. If such a \(k\) exists, \(a\) is said to have finite order \(k\). If no such \(k\) exists, \(a\) is said to have infinite order. The order of an element gives us a way to measure how elements in a group interact through the operation defined by the group. Knowing the order is crucial when exploring properties of groups and understanding how they behave in conjunction with other elements. It determines the "cycle" or the periodic nature of the element within the group.

Least Common Multiple

The **least common multiple** (LCM) of two integers is the smallest positive integer that is a multiple of each integer. When dealing with group theory, the concept of LCM is used to determine the order of an element in a product group. For two numbers \(m\) and \(n\), their LCM, denoted as \(\operatorname{lcm}(m, n)\), is calculated using the formula:

• \(\operatorname{lcm}(m, n) = \frac{m \cdot n}{\gcd(m, n)}\).

This formula divides the product of the two numbers by their greatest common divisor (GCD), ensuring the LCM is not unnecessarily large. The LCM helps in finding the periodic return of simultaneous cycles in sequences, akin to finding when two clock hands meet.

Product Group

A **product group** is a mathematical structure that combines two groups, \(G\) and \(H\), into a single group \(G \times H\), consisting of ordered pairs \((g, h)\), where \(g \in G\) and \(h \in H\). The operation in a product group is defined component-wise, meaning if you have two elements \((g_1, h_1)\) and \((g_2, h_2)\), their product is \((g_1g_2, h_1h_2)\).The order of an element \((c, d)\) in \(G \times H\) is determined by the least common multiple of the orders of \(c\) in \(G\) and \(d\) in \(H\). This is because each component of the pair operates independently based on its group conditions. Understanding the structure of product groups allows mathematicians to combine the properties of multiple groups and explore their interactions.

Greatest Common Divisor

The **greatest common divisor** (GCD) of two numbers is the largest positive integer that divides each of the numbers without leaving a remainder. This concept is essential in various calculations, particularly in finding the least common multiple. If two numbers have a common factor greater than one, like \(q\) in this exercise, it means they share a divisor. The GCD is utilized to simplify ratios and fractions in many mathematical problems. By calculating the GCD, it allows for the simplification of expressions where the least common multiple is concerned, such as in finding the order of elements in product groups. This relationship through division ensures efficient computation and deeper understanding of divisors and multiples.