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2.15. (a) Prove that GL2 (Fp) is a group. (b) Show that GL2 (Fp) is a noncommutative group for every prime p. (c) Describe GL2 (F2) completely. That is, list its elements and describe the multi- plication table. (d) How many elements are there in the group GL2(Fp)? (e) How many elements are there in the group GLn (Fp)?

2.15. (a) Prove that GL2 (Fp) is a group. (b) Show that GL2 (Fp) is a noncommutative group for every prime p. (c) Describe GL2 (F2) completely. That is, list its elements and describe the multi- plication table. (d) How many elements are there in the group GL2(Fp)? (e) How many elements are there in the group GLn (Fp)?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Chapter 2: Linear Algebra - Group Theory #### Exercise 2.15 This exercise set explores the properties and structure of the general linear group of 2x2 invertible matrices over a finite field, denoted as \( GL_2(\mathbb{F}_p) \). **(a) Prove that \( GL_2(\mathbb{F}_p) \) is a group.** This question requires you to demonstrate that the set of all 2x2 invertible matrices over the finite field \( \mathbb{F}_p \) satisfies the group axioms: 1. **Closure**: The product of any two invertible matrices is also invertible. 2. **Associativity**: Matrix multiplication is associative. 3. **Identity Element**: There exists an identity matrix \( I \) such that for any matrix \( A \in GL_2(\mathbb{F}_p) \), \( AI = IA = A \). 4. **Inverse Element**: Every matrix \( A \in GL_2(\mathbb{F}_p) \) has an inverse \( A^{-1} \) such that \( AA^{-1} = A^{-1}A = I \). **(b) Show that \( GL_2(\mathbb{F}_p) \) is a noncommutative group for every prime \( p \).** This part asks you to demonstrate that for any prime \( p \), matrix multiplication in \( GL_2(\mathbb{F}_p) \) is generally not commutative; i.e., there exist matrices \( A \) and \( B \) in \( GL_2(\mathbb{F}_p) \) such that \( AB \neq BA \). **(c) Describe \( GL_2(\mathbb{F}_2) \) completely. That is, list its elements and describe the multiplication table.** Here, you are asked to detail the specific elements of \( GL_2(\mathbb{F}_2) \) and provide a multiplication table, showing how every pair of elements multiplies. **(d) How many elements are there in the group \( GL_2(\mathbb{F}_p) \)?** This question involves counting the number of invertible 2x2 matrices over \( \mathbb{F}_p \). To solve this, you must consider
Transcribed Image Text:### Chapter 2: Linear Algebra - Group Theory #### Exercise 2.15 This exercise set explores the properties and structure of the general linear group of 2x2 invertible matrices over a finite field, denoted as \( GL_2(\mathbb{F}_p) \). **(a) Prove that \( GL_2(\mathbb{F}_p) \) is a group.** This question requires you to demonstrate that the set of all 2x2 invertible matrices over the finite field \( \mathbb{F}_p \) satisfies the group axioms: 1. **Closure**: The product of any two invertible matrices is also invertible. 2. **Associativity**: Matrix multiplication is associative. 3. **Identity Element**: There exists an identity matrix \( I \) such that for any matrix \( A \in GL_2(\mathbb{F}_p) \), \( AI = IA = A \). 4. **Inverse Element**: Every matrix \( A \in GL_2(\mathbb{F}_p) \) has an inverse \( A^{-1} \) such that \( AA^{-1} = A^{-1}A = I \). **(b) Show that \( GL_2(\mathbb{F}_p) \) is a noncommutative group for every prime \( p \).** This part asks you to demonstrate that for any prime \( p \), matrix multiplication in \( GL_2(\mathbb{F}_p) \) is generally not commutative; i.e., there exist matrices \( A \) and \( B \) in \( GL_2(\mathbb{F}_p) \) such that \( AB \neq BA \). **(c) Describe \( GL_2(\mathbb{F}_2) \) completely. That is, list its elements and describe the multiplication table.** Here, you are asked to detail the specific elements of \( GL_2(\mathbb{F}_2) \) and provide a multiplication table, showing how every pair of elements multiplies. **(d) How many elements are there in the group \( GL_2(\mathbb{F}_p) \)?** This question involves counting the number of invertible 2x2 matrices over \( \mathbb{F}_p \). To solve this, you must consider
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