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 , which is the group of integers modulo 10 under addition.
Step 1: Automorphism group of
The group 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 are elements that generate all other elements of the group under addition modulo 10. These generators are the integers in that are relatively prime to 10 (i.e., their greatest common divisor (GCD) with 10 is 1).
Generators of :
The elements of are . Among these, the elements relatively prime to 10 are: Thus, the generators of are .
Step 2: Automorphism action
An automorphism of is a bijective homomorphism. Because is cyclic, every automorphism is determined by where it maps the generator .
If , where , then is a valid automorphism if and only if is a generator of . From Step 1, we know that .
Step 3: Valid values of
The possible values of such that is an automorphism are:
Final Answer:
The possible values of are:
Let me know if you'd like further clarification! Here are some related questions to expand on this concept:
- What is the definition of an automorphism of a group?
- How do you find the generators of any cyclic group ?
- Can you calculate the automorphism group of ?
- What is the structure of the automorphism group of for any ?
- How does the Euler totient function relate to the number of automorphisms of ?
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