Problem 6 - A Book of Abstract Algebra

Question

For any positive integer \(m\), what is the index of \(\langle m\rangle\) in \(\mathbb{Z}\) ?

Step-by-Step Solution

Answer

The index of \( \langle m \rangle \) in \( \mathbb{Z} \) is \( m \).

Step 1: Understanding the Problem

We want to find the index of the subgroup \( \langle m \rangle \) in the group of integers \( \mathbb{Z} \), where \( m \) is a positive integer. The notation \( \langle m \rangle \) represents the subgroup generated by \( m \), which consists of all multiples of \( m \).

Step 2: Definition of Index

The index of a subgroup \( H \) in a group \( G \), denoted \( [G : H] \), is defined as the number of distinct left cosets of \( H \) in \( G \). For our problem, \( H = \langle m \rangle \) and \( G = \mathbb{Z} \).

Step 3: Finding Cosets of \( \langle m \rangle \)

The left cosets of \( \langle m \rangle \) in \( \mathbb{Z} \) are of the form \( n + \langle m \rangle \) where \( n \in \mathbb{Z} \). Each coset is the set of all integers of the form \( n + mk \) where \( k \in \mathbb{Z} \).

Step 4: Counting the Distinct Cosets

For \( n + \langle m \rangle \) and \( n' + \langle m \rangle \) to be the same coset, the two must differ by an element of \( \langle m \rangle \), which happens if \( n - n' = km \) for some integer \( k \). This implies that \( n \equiv n' \pmod{m} \).

Step 5: Calculating the Index

The number of distinct cosets (or equivalently, the number of different possible remainders when dividing by \( m \)) is exactly \( m \). Therefore, the index of \( \langle m \rangle \) in \( \mathbb{Z} \) is \( m \).

Step 6: Conclusion

Thus, the index of \( \langle m \rangle \) in \( \mathbb{Z} \) is the number of elements in the complete residue system modulo \( m \), which is \( m \).

Key Concepts

Subgroup

In group theory, a subgroup is a smaller group contained within a larger group that itself satisfies the group properties. Here's how it works:

• A subgroup must contain the identity element of the larger group.
• For any two elements within the subgroup, the product of these elements must also be in the subgroup.
• Every element in the subgroup must have an inverse within the same subgroup.

In our exercise, the subgroup \( \langle m \rangle \) is generated by the integer \( m \), meaning it consists of all multiples of \( m \) within the set of integers \( \mathbb{Z} \). This subgroup is a neat example because it visually depicts the repetition and predictability often found in group structures.

Coset

A coset is a way to partition a group into pieces or sections that have similar structure. Here's a breakdown:

• Left cosets are created by multiplying a fixed element on the left of each element in the subgroup.
• Each coset is essentially a shift of the subgroup by a certain amount.
• In our problem, the cosets of \( \langle m \rangle \) in \( \mathbb{Z} \) are of the form \( n + \langle m \rangle \), which means adding \( n \) to every multiple of \( m \).

To simplify, think of cosets as copies of the subgroup that can "slide around" the group without changing their core structure. This sliding by adding \( n \) does not alter the basic repeating pattern of the subgroup's multiples of \( m \).

Index of a Subgroup

The index of a subgroup within a group is incredibly useful for understanding how the subgroup divides the group. Specifically, it is the number of distinct cosets of the subgroup within the group.

• Mathematically, the index is written as \([G : H]\) where \(G\) is the group and \(H\) is the subgroup.
• It tells us how many times a subgroup fits into the larger group.
• In the integer group \( \mathbb{Z} \), the index of \( \langle m \rangle \) is found by counting how many integer cosets fall into one cycle or complete set of residues modulo \(m\) – essentially, it's \(m\).

Understanding the index allows us to see the relationship between the size of the subgroup and the size of the entire group, casting light on the structure and symmetry of the mathematical universe of integers as divided by \(m\).

Integer Group \(\mathbb{Z}\)

The integer group, denoted by \(\mathbb{Z}\), is a fundamental and infinite group in mathematics. Here's what you should know:

• It includes all whole numbers, both positive and negative, including zero.
• The group operation here is addition, making \(\mathbb{Z}\) an additive group.
• It is a straightforward example of an infinite group, used often in abstract algebra and number theory.

In our exercise, \(\mathbb{Z}\) serves as the parent group in which the subgroup \(\langle m \rangle\) exists. Integer arithmetic forms the backbone of many mathematical areas, demonstrating both structure and endless possibility. This abundance of integers provides a rich playground for exploring how different mathematical objects, like subgroups and cosets, interact.