Math Problem Statement

Let  be an element of the automorphism group of ℤ10. Then, find the possible values of ) 9 1 (  k k such that ] [ ])

Solution

To solve the problem, let's carefully analyze the given situation. The problem involves an automorphism of the group Z10\mathbb{Z}_{10}, which is the group of integers modulo 10 under addition.


Step 1: Automorphism group of Z10\mathbb{Z}_{10}

The group Z10\mathbb{Z}_{10} is a cyclic group of order 10. For an automorphism of a cyclic group, the key property is that the automorphism maps generators of the group to generators of the group.

The generators of Z10\mathbb{Z}_{10} are elements that generate all other elements of the group under addition modulo 10. These generators are the integers in Z10\mathbb{Z}_{10} that are relatively prime to 10 (i.e., their greatest common divisor (GCD) with 10 is 1).

Generators of Z10\mathbb{Z}_{10}:

The elements of Z10\mathbb{Z}_{10} are {0,1,2,3,4,5,6,7,8,9}\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}. Among these, the elements relatively prime to 10 are: {1,3,7,9}.\{1, 3, 7, 9\}. Thus, the generators of Z10\mathbb{Z}_{10} are {1,3,7,9}\{1, 3, 7, 9\}.


Step 2: Automorphism action

An automorphism α\alpha of Z10\mathbb{Z}_{10} is a bijective homomorphism. Because Z10\mathbb{Z}_{10} is cyclic, every automorphism is determined by where it maps the generator 11.

If α(1)=k\alpha(1) = k, where 1k91 \leq k \leq 9, then α\alpha is a valid automorphism if and only if kk is a generator of Z10\mathbb{Z}_{10}. From Step 1, we know that k{1,3,7,9}k \in \{1, 3, 7, 9\}.


Step 3: Valid values of kk

The possible values of kk such that α\alpha is an automorphism are: k{1,3,7,9}.k \in \{1, 3, 7, 9\}.


Final Answer:

The possible values of kk are: k=1,3,7,9.k = 1, 3, 7, 9.


Let me know if you'd like further clarification! Here are some related questions to expand on this concept:

  1. What is the definition of an automorphism of a group?
  2. How do you find the generators of any cyclic group Zn\mathbb{Z}_n?
  3. Can you calculate the automorphism group of Z12\mathbb{Z}_{12}?
  4. What is the structure of the automorphism group of Zn\mathbb{Z}_n for any nn?
  5. How does the Euler totient function ϕ(n)\phi(n) relate to the number of automorphisms of Zn\mathbb{Z}_n?

Tip: When solving automorphism problems for cyclic groups, remember that automorphisms are uniquely determined by the images of generators!

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Math Problem Analysis

Mathematical Concepts

Group Theory
Cyclic Groups
Automorphisms
Relatively Prime Numbers

Formulas

Generators of a cyclic group are elements relatively prime to the group order
Automorphism property: α(g) must be a generator if g is a generator

Theorems

Structure of Cyclic Groups
Automorphisms of Cyclic Groups

Suitable Grade Level

Undergraduate - Abstract Algebra