91Ó°ÊÓ

An integer matrix is a matrix where all the elements are integers. In this problem, we consider a specific kind of integer matrix which represents a subgroup of the group \( \mathbb{Z}^2 \).

The matrix given is \( A = \begin{pmatrix} a & b \ 0 & d \end{pmatrix} \), where \( a \) and \( d \) are positive integers. These integers satisfy the condition \( ad = n \), meaning the product of \( a \) and \( d \) equals \( n \), the index of the subgroup.

• The integer \( a \) corresponds to one dimension of the subgroup structure.
• The integer \( d \) corresponds to the second dimension.
• \( b \) can take any integer value such that \( 0 \leq b < d \).

This specific form helps in counting the number of distinct finite index subgroups in \( \mathbb{Z}^2 \). Understanding how integer matrices represent these subgroups is crucial for solving the given problem.

Divisor pairs are a key element to solving the subgroup problem in \( \mathbb{Z}^2 \). Given an integer \( n \), its divisors are integers that divide \( n \) without leaving a remainder.

For any divisor \( a \) of \( n \), you can find a corresponding divisor \( d \) such that \( ad = n \). This pairing, \( (a, d) \), provides the foundational structure for the matrix \( A \) in the problem.

• Each divisor \( a \) implies the existence of \( d = \frac{n}{a} \).
• The divisor pairs \( (a, d) \) are used to build the integer matrices that represent subgroups of \( \mathbb{Z}^2 \).

By calculating all such pairs, we know how \( n \) can decompose into two numbers, guiding us to the number of possible subgroups of the finite index \( n \).

The divisor sum function, denoted as \( \sigma(n) \), is a function that returns the sum of all positive divisors of \( n \).

In the context of the problem, \( \sigma(n) \) is directly related to the number of subgroups of a given index in \( \mathbb{Z}^2 \). This relationship arises because:

• For each divisor \( a \) of \( n \), there are \( \frac{n}{a} \) valid choices for \( b \) in the matrix \( A \).
• The total number of subgroups is thus the sum over all divisors \( a \), of the expression \( \frac{n}{a} \).

The mathematical expression \( \sum_{a \mid n} \frac{n}{a} \) matches exactly with \( \sigma(n) \), thereby proving that there are \( \sigma(n) \) such subgroups.
The divisor sum function is a powerful tool used in number theory, often simplifying complex counting into more manageable forms.

Combinatorial arguments provide a method of counting discrete structures, which in this case, are the subgroups of a given index in \( \mathbb{Z}^2 \).

The process involves a few key steps:

• Identify divisor pairs \( (a, d) \), ensuring \( ad = n \).
• For each pair, determine potential values for \( b \), constrained by the range \( 0 \leq b < d \).

These steps ensure you account for all possible matrices \( A \) that form subgroups of \( \mathbb{Z}^2 \) with index \( n \). The product of choices for \( a, d, \) and \( b \) leads to the realization that the count is \( \sigma(n) \).
In more general terms, similar arguments can extend to higher dimensions, such as \( \mathbb{Z}^k \), resulting in \( \sigma(n)^k \), where each additional dimension complicates the counting procedure. Combinatorial reasoning simplifies these calculations into understandable and logical steps.