Problem 8 - Algebra: Pure and Applied

Question

Let \(G=\langle a\rangle\) be a cyclic group of order \(20 .\) Find all the elements \(b \in G\) of order \(|b|=10\).

Step-by-Step Solution

Answer

The elements of order 10 are \(a^2, a^6, a^{14}, a^{18}\).

Step 1: Understand the structure of the group

Since G is a cyclic group of order 20, it is generated by the element \(a\). Thus, every element of \(G\) can be expressed in the form \(a^k\) where \(k\) is an integer between 0 and 19 (inclusive).

Step 2: Determine the condition for order of an element

The order of an element \(b = a^k\) in \(G\) is the smallest positive integer \(m\) such that \(b^m = e\), where \(e\) is the identity element of the group. For \(b = a^k\), this means \(a^{km} = e = a^{20}\), so \(km\) must be a multiple of 20. The order \(|b|\) or \(m\) is therefore \(\frac{20}{\text{gcd}(k, 20)}\).

Step 3: Apply condition for order 10

We need \(|b| = 10\), so \(\frac{20}{\text{gcd}(k, 20)} = 10\). This implies that \(\text{gcd}(k, 20) = 2\).

Step 4: Find integers \(k\) such that \(\text{gcd}(k, 20) = 2\)

Find all integers \(k\) in \([0, 19]\) such that \(\text{gcd}(k, 20) = 2\). These integers satisfy \(k = 2, 6, 14, 18\).

Step 5: Verify the elements

Each \(k\) gives an element \(b = a^k\) with \(\text{gcd}(k, 20) = 2\), meaning each element \(a^2, a^6, a^{14}, a^{18}\) has order 10. Thus, these are all the elements of \(G\) that have order 10.

Key Concepts

Order of an Element

The order of an element in a group is a fundamental concept in group theory. In a group, the order of an element is the smallest positive integer, say \( m \), for which raising the element to this power results in the identity element of the group. In mathematical terms, for an element \( b \) of a group \( G \), the order \( |b| \) is the smallest \( m \) such that \( b^m = e \), where \( e \) denotes the identity element of the group.

To find the order of an element \( b \) in a cyclic group of order 20, where \( b = a^k \), we check when \( (a^k)^m = e \). This simplifies to \( a^{km} = a^{20} \) because the group order is 20. Therefore, \( km \) has to be a multiple of the group order (20). Understanding the order of an element is crucial for determining the structure of the elements within cyclic groups.

gcd (Greatest Common Divisor)

The greatest common divisor, or gcd, is a key concept in mathematics used to determine the largest number that divides two or more numbers without leaving a remainder.

In the context of finding elements in a cyclic group \( G \) with a specific order, we use the gcd to find when certain powers of the generator result in the desired group element order. For example, if we want the order of \( a^k \) to be 10, we set \( \frac{20}{\text{gcd}(k, 20)} = 10 \). This equation directly relates the gcd of \( k \) and the group order (20) to the desired order of \( a^k \).

• If \( \text{gcd}(k, 20) = 2 \), then the element \( a^k \) will have order 10.

Using the gcd helps us to deduce which power values of the generator will yield elements with particular properties within the group.

Identity Element

The identity element is a vital part of any group. It is the element that leaves other elements unchanged when combined with them according to the group operation. If \( G \) is a group, and \( e \) is the identity element, then for every element \( a \) in \( G \), the relation \( a \cdot e = e \cdot a = a \) holds true.

In cyclic groups, the identity element is especially important because it is the point where powers of the generator "restart." For a cyclic group \( G = \langle a \rangle \) of order 20, \( e = a^{20} \). This means that any element \( a^k \) raised to the proper power will return to the identity element, reinforcing the cyclic structure.

Group Theory

Group theory is the branch of mathematics that studies the algebraic structures known as groups. Groups consist of a set of elements together with an operation that combines any two elements to form a third element. This operation must satisfy four key properties:

• Closure: Performing the operation on any two elements of the group results in another element still within the group.
• Associativity: The grouping of elements, when combined, does not affect the outcome.
• Identity: There exists an element (identity element) in the group which, when used in the operation with any group element, returns the element itself.
• Inverses: Every element has an inverse, such that operating the element and its inverse yields the identity element.

Understanding these properties aids in solving problems involving cyclic groups, like finding elements of specific order. In the original exercise, since \( G \) is cyclic with a generator \( a \), each element of \( G \) can be expressed as a power of \( a \). Group theory helps formalize how these elements interact and determine their order based on the group structure.