Problem 40 - A First Course in Abstract Algebra

Question

Find the order of the torsion subgroup of \(\mathrm{Z}_{4} \times \mathbb{Z} \times \mathrm{Z}_{1}\); of \(\mathrm{Z}_{12} \times \mathrm{Z} \times \mathrm{Z}_{12}\).

Step-by-Step Solution

Answer

The order of the torsion subgroup is 4 for \( \mathrm{Z}_{4} \times \mathbb{Z} \times \mathrm{Z}_{1} \), and 144 for \( \mathrm{Z}_{12} \times \mathbb{Z} \times \mathrm{Z}_{12} \).

Step 1: Identify torsion elements in the given groups

The torsion subgroup of an abelian group consists of all elements that have finite order. In the group \( \mathrm{Z}_{4} \times \mathbb{Z} \times \mathrm{Z}_{1} \), the elements from \( \mathbb{Z} \) are not torsion because they have infinite order, and elements from \( \mathrm{Z}_{1} \) are the identity element. Thus, torsion elements come only from \( \mathrm{Z}_{4} \). In the group \( \mathrm{Z}_{12} \times \mathbb{Z} \times \mathrm{Z}_{12} \), similar reasoning applies: only elements from \( \mathrm{Z}_{12} \times \mathrm{Z}_{12} \) are torsion, as elements from \( \mathbb{Z} \) have infinite order.

Step 2: Find the torsion subgroup

For \( \mathrm{Z}_{4} \times \mathbb{Z} \times \mathrm{Z}_{1} \), the torsion subgroup is \( \mathrm{Z}_{4} \), as order-4 elements and identity contribute. For \( \mathrm{Z}_{12} \times \mathbb{Z} \times \mathrm{Z}_{12} \), the torsion subgroup is \( \mathrm{Z}_{12} \times \mathrm{Z}_{12} \) because each element in either \( \mathrm{Z}_{12} \) has finite order.

Step 3: Calculate the order of the torsion subgroup

The order of the torsion subgroup in \( \mathrm{Z}_{4} \times \mathbb{Z} \times \mathrm{Z}_{1} \) is \( 4 \), since it includes all elements of \( \mathrm{Z}_{4} \). For \( \mathrm{Z}_{12} \times \mathbb{Z} \times \mathrm{Z}_{12} \), the order is \( 144 \), which is the product of the orders of the two \( \mathrm{Z}_{12} \) subgroups, i.e., \( 12 \times 12 \).

Key Concepts

Abelian Group

An abelian group is a mathematical structure where the group operation is commutative. This means that the order in which you perform the group operation on elements does not matter. For a group \( G \), this property is expressed as \( a \cdot b = b \cdot a \) for all \( a, b \) in \( G \).

Abelian groups are named after the mathematician Niels Henrik Abel. They have an important feature that simplifies many mathematical explorations:

• Closure: the group operation on any two elements will result in another element that is still within the group.
• Associativity: the grouping of operations doesn’t affect the result.
• Identity: there is a special element that works as a neutral result for the group operation.
• Inverses: every element has another element that "undoes" the operation with it, returning to the identity element.
• Commutative property: as mentioned earlier, the order of applying the operation doesn’t change the outcome.

Understanding these properties can provide clarity when dealing with more complex structures such as torsion subgroups.

Finite Order

In group theory, an element of a group is said to have finite order if applying the group operation to the element a certain number of times leads back to the identity element. This means that the element "cycles" back to start after a finite number of steps. The smallest positive number \( n \) such that \( a^n = e \), where \( e \) is the identity element, is called the order of the element.

Here's how it works:

• An element with finite order can be visualized as tracing a regular, repeating path through the group until it loops back to the beginning.
• If \( a \) is in a group \( G \), and \( a^n = e \), then \( a \) has finite order \( n \). The power \( n \) signifies how many times you apply the group operation to loop back to the identity element.

This concept plays a critical role in understanding torsion subgroups, as torsion involves elements of finite order.

Group Theory

Group theory is a branch of mathematics dealing with groups, which are sets equipped with an operation that combines any two elements to form a third. It is an abstract way to capture the essence of symmetry and provides handy tools for various fields including physics, chemistry, and computer science.

Key ideas in group theory include:

• Elements and Operations: Understand that each group consists of elements and a binary operation that combines them.
• Identity and Inverses: Recognize the importance of a 'do-nothing' element (identity) and the idea that every element has an opposite (inverse).
• Cyclic and Infinite Groups: Be aware of differences between groups where elements repeat regularly (cyclic) versus continue indefinitely (infinite).

Group theory builds a framework to analyze and solve problems involving symmetry and transformations. The application to torsion subgroups digs deeper by focusing on elements that possess finite order properties.