Question

Let \(G=\langle a\rangle\) be a cyclic group of order \(10 .\) Describe explicitly the elements of \(\operatorname{Aut}(G)\).

Short answer

\( \operatorname{Aut}(G) = \{\phi_1, \phi_3, \phi_7, \phi_9\} \), isomorphic to \( \mathbb{Z}_4 \).

Step-by-step solution

Understanding Aut(G)

The automorphism group, \( \operatorname{Aut}(G) \), of a cyclic group \( G \) of order \( n \) consists of all isomorphisms from \( G \) onto itself. These automorphisms are determined by where they send a generator of \( G \).

Identify Cyclic Group Elements

Since \( G = \langle a \rangle \) is a cyclic group of order \( 10 \), the elements of \( G \) are \( \{ e, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9 \} \), where \( e \) is the identity element.

Determine Valid Automorphisms

Any automorphism of \( G \) must map \( a \) to a generator of \( G \). The generators of \( G \) are those elements such that their order is \( 10 \). These elements in \( G \) are \( a^1, a^3, a^7, \) and \( a^9 \).

Construct Automorphisms

Each generator can correspond to a unique automorphism. For each \( a^k \) which is a generator, define an automorphism \( \phi_k : G \to G \) by \( \phi_k(a) = a^k \). This gives us the mappings: 1. \( \phi_1(a) = a \) (identity map) 2. \( \phi_3(a) = a^3 \) 3. \( \phi_7(a) = a^7 \) 4. \( \phi_9(a) = a^9 \)

Verify Distinct Automorphisms

Verify that each of these mappings defines a distinct and valid automorphism. Each maps \( a \) to a different generator of \( G \), and preserves group operation as required by an isomorphism. Thus, these mappings: \( \phi_1, \phi_3, \phi_7, \phi_9 \) are distinct automorphisms.

Determine Structure of Aut(G)

Since there are four automorphisms, \( \operatorname{Aut}(G) \) is isomorphic to the group of units of the integers modulo 10, denoted \( \mathbb{Z}_{10}^* \), which is of order 4. It is isomorphic to the cyclic group \( \mathbb{Z}_4 \).

Key concepts

Cyclic Groups

Cyclic groups are essential building blocks in group theory. These are algebraic structures where you can generate every element of the group by repeatedly applying the group operation to a particular element, known as the generator. A cyclic group, denoted as \( G = \langle a \rangle \), indicates that every element of \( G \) can be expressed as some power of \( a \). For instance, if \( G \) is a cyclic group of order 10, then \( G \) consists of precisely 10 elements: \( \{ e, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9 \} \), where \( e \) represents the identity element. These groups are notable for their simplicity and structure.
The generator \( a \) plays a crucial role since every operation within \( G \) can be traced back to it. This property makes cyclic groups easily understandable and a fundamental concept in various areas of mathematics, including automorphism groups, number theory, and symmetry operations.

Group Theory

Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group consists of a set equipped with an operation that combines any two elements to form a third element, satisfying the conditions of closure, associativity, identity, and invertibility.
The study of groups includes understanding their structure, classification, and how they can transform geometrical objects or solve algebraic equations. Groups are classified into different types, with cyclic groups being one of the simplest forms. They are pivotal in understanding more complex structures through their fundamental nature.
Within group theory, the concept of automorphisms, which are isomorphisms from a group onto itself, plays an important role. For any group \( G \), the set of all automorphisms forms a group, known as the automorphism group of \( G \), noted as \( \operatorname{Aut}(G) \). In cyclic groups, these automorphisms can be explicitly determined and provide insight into the symmetry and transformation properties of the group.

Generators in Groups

The concept of generators in groups highlights how certain elements can produce the entire group's structure through repeated operations. In a cyclic group \( G = \langle a \rangle \), the generator \( a \) is key because every element in the group is a power of this generator.
Different elements might serve as generators if they can recreate all elements in the group via their powers. For our example with the cyclic group \( G \) of order 10, elements like \( a^1, a^3, a^7, \) and \( a^9 \) each generate the group because the only requirement for a generator is that it has the same order as the group itself.
Understanding which elements are generators helps in determining valid automorphisms, which map a generator to another generator while preserving the group's operation. This ability to map within the group structure is foundational in defining and exploring automorphism groups.