Problem 4

Question

Let $p$ be a prime, and let $m$ be a natural number. The kernel of the natural homomorphism $$ G L\left(2, Z / p^{m} Z\right) \longrightarrow G \mathrm{~L}\left(2, \mathbb{Z} / p^{m-1} \mathbb{Z}\right) $$ is isomorphic to the additive group of all $2 \times 2$ matrices with entries in $\mathbb{Z} / p \mathbb{Z}$. Using this, show: $$ \begin{aligned} &\\# G L\left(2, Z / p^{m}{Z}\right)=p^{4 m-3}\left(p^{2}-1\right)(p-1) \\ &\\# S L\left(2, \mathbb{Z} / p^{m} \mathcal{Z}\right)=p^{3 m-2}\left(p^{2}-1\right) \end{aligned} $$

Step-by-Step Solution

Verified

Answer

The specified group orders are derived using kernel properties and canonical group calculations in modular arithmetic.

1Step 1: Understanding the Kernel

The kernel of a homomorphism between two groups is the set of elements that map to the identity element of the codomain. In our case, the kernel is isomorphic to the additive group of all 2x2 matrices with entries in $\mathbb{Z}/p\mathbb{Z}$. This means there are $p^4$ possible matrices in the kernel as there are four entries each with $p$ choices.

2Step 2: The Exact Sequence of Inclusion

Given a short exact sequence induced by the natural homomorphism, the order of $GL(2, \mathbb{Z}/p^m\mathbb{Z})$ can be iteratively calculated by multiplying the order of $\ker$ and the order of $GL(2, \mathbb{Z}/p^{m-1}\mathbb{Z})$.

3Step 3: Calculate the Order of GL(2, \mathbb{Z}/p\mathbb{Z})

First find the order of $GL(2, \mathbb{Z}/p\mathbb{Z})$. The set of 2x2 invertible matrices over $\mathbb{Z}/p\mathbb{Z}$ has $(p^2 - 1) \times (p^2 - p) = (p^2 - 1)(p - 1)$ elements. The first entry can be any of $p^2 - 1$ values (non-zero determinant), and the second entry can differ from the first by $p$ values.

4Step 4: Order of GL(2, \mathbb{Z}/p^m\mathbb{Z})

Using induction or the multiplication principle from group theory, the order of $GL(2, \mathbb{Z}/p^m\mathbb{Z})$ is obtained by multiplying the previous step’s order with $p^4$ repeatedly as $m$ increases, obtaining $p^{4m-3}(p^2-1)(p-1)$ for its order.

5Step 5: The Order of SL(2, \mathbb{Z}/p^m\mathbb{Z})

SL(2, \mathbb{Z}/p^m\mathbb{Z}) is a subgroup of GL(2, \mathbb{Z}/p^m\mathbb{Z}) consisting of matrices with determinant 1. Using the previously determined order of GL and noting the group properties, we find that the order of SL is $p^{3m-2}(p^2-1)$. This is achieved by adjusting the GL order according to subgroup index principles.

Key Concepts

GL(2, $\mathbb{Z}/p^m\mathbb{Z}$)Kernel of HomomorphismOrder of Groups

GL(2, $\mathbb{Z}/p^m\mathbb{Z}$)

The notation $GL(2, \mathbb{Z}/p^m\mathbb{Z})$ represents the general linear group of invertible 2x2 matrices with entries from the ring $\mathbb{Z}/p^m\mathbb{Z}$. This group plays a critical role in understanding transformations and symmetries in a modular arithmetic setting.

The matrices in $GL(2, \mathbb{Z}/p^m\mathbb{Z})$ must have a non-zero determinant when calculated under modular arithmetic. The process to determine the order - meaning the number of elements - of this group involves a careful analysis of choices for each matrix entry that ensures the determinant is indeed invertible.

For a simple case like $m = 1$, i.e., $\mathbb{Z}/p\mathbb{Z}$, there are $p^2 - 1$ possibilities for the first row, ensuring the first entry isn't zero.
The second row must be linearly independent, offering $p^2 - p$ choices.

Consequently, the order of $GL(2, \mathbb{Z}/p\mathbb{Z})$ is $(p^2 - 1)(p - 1)$. As $m$ increases, the order is computed recursively using the modification by the kernel contribution, leading to the formula $p^{4m-3}(p^2-1)(p-1)$.

Kernel of Homomorphism

In group theory, the kernel of a homomorphism is a fundamental concept. It consists of all elements in the domain group that map to the identity element of the codomain group through the homomorphism function.

For the linear group homomorphism $GL(2, \mathbb{Z}/p^m\mathbb{Z}) \rightarrow GL(2, \mathbb{Z}/p^{m-1}\mathbb{Z})$, the kernel is isomorphic to the group of all 2x2 matrices with coefficients in $\mathbb{Z}/p\mathbb{Z}$ that contribute no real transformation in the homomorphic image.

This means any matrix in the kernel, when considered under the homomorphism, results in the zero matrix or identity effect in the codomain.
There are precisely $p^4$ such matrices since each entry has $p$ possible values.

This kernel's structure provides a foundation for understanding the step-by-step order computation of the larger group and its subgroups.

Order of Groups

The concept of order in groups describes the total number of elements the group contains. When determining the order of $GL(2, \mathbb{Z}/p^m\mathbb{Z})$, a pivotal step involves recognizing the contribution of different group sequences and kernel integration.

The process starts with $GL(2, \mathbb{Z}/p\mathbb{Z})$, as understanding this base instance helps scale to more complex cases as $m$ increases. Group order is built incrementally by:

Initially establishing the order of the base group at $m=1$ using group properties.
Utilizing the kernel's structure precisely $p^4$, housing matrices with negligible transformations from higher powers of $m$.

Multiplying these increments step by step demystifies the intimidating formula for the group's order. Each $m$ step extends the group's complexity, maintaining the logical chain foresaw by the lower orders, culminating in expressions like $p^{4m-3}(p^2-1)(p-1)$.

Problem 4

Problem 5