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Magazine Chapter 2: Problem 10
For \(p\) a prime show that \(\operatorname{Aut}\left(\mathbb{Z}_{p}\right) \cong \mathbb{Z}_{p-1}\).
Short Answer
Expert verified
The automorphism group of \(\mathbb{Z}_{p}\) is isomorphic to \(\mathbb{Z}_{p-1}\) because \(\operatorname{Aut}(\mathbb{Z}_{p}) \cong U(p)\) and \(|U(p)| = p-1\).
Step by step solution
01
Understanding the Problem
We need to show that the group of automorphisms of \([1m\mathbb{Z}_{p}\u001b[0m\), where \([1mp\) is a prime number, is isomorphic to \([1m\mathbb{Z}_{p-1}\u001b[0m\). This means we'll demonstrate a one-to-one correspondence (isomorphism) between these two groups.
02
Define the Automorphism Group
The automorphism group \([1m\operatorname{Aut}\left(\mathbb{Z}_{p}\right)\u001b[0m\) consists of all bijective (invertible) homomorphisms from \([1m\mathbb{Z}_{p}\u001b[0m\) to itself. For a group \([1m\mathbb{Z}_{n}\u001b[0m\), \([1m\operatorname{Aut}\left(\mathbb{Z}_{n}\right)\u001b[0m\) is isomorphic to the group of units \([1mU(n)\u001b[0m\). Thus, \([1m\operatorname{Aut}\left(\mathbb{Z}_{p}\right) = U(p)\u001b[0m\).
03
Identify the Group of Units
The group of units \([1mU(p)\u001b[0m\) in \([1m\mathbb{Z}_{p}\u001b[0m\) is the set of all integers from 1 to \([1mp-1\u001b[0m\) that are coprime to \([1mp\u001b[0m\). Because \([1mp\u001b[0m\) is prime, all integers in this range are coprime to \([1mp\u001b[0m\).
04
Determine the Order of the Group
Since every integer \(1, 2, [1m..., p-1[0m\) is coprime to \([1mp\u001b[0m\), the order of \([1mU(p)\u001b[0m\) is \([1mp-1\u001b[0m\). Therefore, \([1m\operatorname{Aut}\left(\mathbb{Z}_{p}\right)\u001b[0m\) is of order \([1mp-1\u001b[0m\).
05
Show Isomorphism with \(\mathbb{Z}_{p-1}\)
Consider the map \(f: [1mU(p) \to \mathbb{Z}_{p-1}\u001b[0m\) defined by taking the equivalence class of an integer modulo \([1mp-1\u001b[0m\). This map is bijective and respects the group operation, thus establishing the isomorphism \([1m\operatorname{Aut}\left(\mathbb{Z}_{p}\right) \cong \mathbb{Z}_{p-1}\u001b[0m\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Automorphisms
In group theory, an automorphism is a special kind of homomorphism. It is a bijective homomorphism from a group to itself. This means it must preserve the group operation and map every element in the group in a reversible way.
- Automorphisms of a group essentially represent symmetrical transformations of the group structure.
- They highlight the internal symmetries that the group possesses.
Prime Numbers
Prime numbers are integral to number theory and are numbers greater than 1, only divisible by 1 and themselves. Understanding primes is crucial in fields like cryptography and algorithms.
- Primes are the building blocks of natural numbers because every natural number is a product of prime numbers.
- They have a unique role in defining group structures in mathematics.
Group of Units
A group of units, denoted as \( U(n) \), refers to the set of all integers that are coprime to \( n \), where the operation in question is multiplication modulo \( n \). Let's break this down:
- For any number \( n \), its group of units comprises numbers that can combine under multiplication to produce 1, the multiplicative identity.
- In \( \mathbb{Z}_p \), if \( p \) is prime, every number from 1 to \( p-1 \) is included in \( U(p) \).
Isomorphism
In group theory, an isomorphism signifies a profound similarity between two groups, often implying they have the same structure even if they appear different on the surface. Here鈥檚 what it entails:
- An isomorphism is a bijective map that respects the operations of the groups it connects.
- It provides a way to transform problems and solutions from one group context to another with equivalent complexity.
