Problem 4 - A Book of Abstract Algebra

Question

Let \(a\) be an element of order 12 in a group \(G\) Which powers of \(a\) have the same order as \(a\) ? [That is, for what values of \(k\) is \(\left.\operatorname{ord}\left(a^{k}\right)=12 ?\right]\)

Step-by-Step Solution

Answer

The powers of \(a\) with the same order as \(a\) are \(a^1, a^5, a^7, a^{11}.\)

Step 1: Understanding the Problem

We need to determine which powers of an element \(a\) of order 12 have the same order. This means finding integers \(k\) such that \(\operatorname{ord}(a^k) = 12\).

Step 2: Order Condition

The order of an element \(a^k\) is calculated as \(\left.\frac{n}{\gcd(n, k)}\right)\), where \(n=12\) is the order of \(a\) and \(k\) is the integer we're solving for. We want this expression to equal 12.

Step 3: Formulating an Equation

Set \(\frac{12}{\gcd(12, k)} = 12\). For this equation to be true, \(\gcd(12, k)\) must be 1, thus \(a^k\) has order 12 when \(k\) is coprime to 12.

Step 4: Identifying Coprime Numbers

We need to find values of \(k\) such that \(1 \leq k \leq 12\) and \(\gcd(12, k) = 1\). The integers that are coprime with 12 are: 1, 5, 7, and 11.

Step 5: Conclusion and Verification

Verify that for \(k = 1, 5, 7, 11\), the order of \(a^k\) remains 12. Since these values have a gcd of 1 with 12, they are verified to have this property.

Key Concepts

Group Theory

Group theory is a fascinating branch of abstract algebra that studies algebraic structures called "groups." A group is essentially a set equipped with an operation that combines any two elements to form a third element. It satisfies four fundamental properties:

• **Closure:** For any two elements in the group, their product is also in the group.
• **Associativity:** The group operation is associative; that is, for elements \(a, b,\) and \(c,\) the equation \((a \cdot b) \cdot c = a \cdot (b \cdot c)\) holds.
• **Identity Element:** There is an element in the group, called the identity element, that, when combined with any element \(a,\) leaves \(a\) unchanged.
• **Inverse Element:** For every element \(a\) in the group, there exists an element \(b\) (the inverse of \(a\)) such that \(a \cdot b\) is the identity element.

Groups can be finite or infinite, depending on the number of elements they contain. Understanding group theory is fundamental for studying more complex algebraic systems.

Order of an Element

The order of an element in a group is an essential concept in group theory. It refers to the smallest positive integer \(n\) such that \(a^n = e,\) where \(e\) is the identity element of the group. Intuitively, it tells us how many times we need to apply the group operation to the element \(a\) to get back to the identity.
In our exercise, the element \(a\) has an order of 12, meaning that if we performed the group operation with \(a\) repeatedly 12 times, we would return to the starting point, the identity.
Finding powers of \(a\) where \(a^k\) retains the same order requires that \(k\) and the order of \(a\) (which is 12) be coprime integers.

Coprime Integers

Coprime integers, or relatively prime numbers, are two numbers that have no positive divisors in common other than 1. In simpler terms, if the greatest common divisor (GCD) of two numbers is 1, they are considered coprime.
In the context of our group theory exercise, we are interested in finding values of \(k\) that are coprime to 12. The goal is to ensure that \(a^k\) has the same order as \(a,\) which is 12. This is achieved when the greatest common divisor \(\gcd(12, k) = 1.\)
The numbers 1, 5, 7, and 11 fit our criteria when we look at numbers up to 12. These integers do not share any prime factors with 12, making them the only values of \(k\) that ensure the order of \(a^k\) is equivalent to 12.

Greatest Common Divisor GCD

The greatest common divisor (GCD) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. The concept of GCD is not only fundamental in number theory but also in understanding the behavior of group elements.
In order to determine whether \(a^k\) will have the same order as \(a,\) we employed the use of GCD in our calculations. The condition required was that \(\frac{12}{\gcd(12,k)} = 12,\) leading to the necessity of \(\gcd(12, k) = 1.\)
Calculating the GCD of small integers typically involves listing out the divisors of the numbers, but for larger calculations, algorithms such as the Euclidean algorithm are paper-efficient and computationally feasible methods to help find the GCD.