What is a group in group theory?

A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for every element.

What are the main properties of a group?

The four main properties are: Closure: The result of the operation on any two elements of the group is also in the group. Associativity: The group operation is associative. Identity: There exists an element that does not change other elements when used in the operation. Invertibility: Every element has an inverse that, when combined with the element, yields the identity.

What is the identity element in a group?

The identity element is a unique element in the group that, when combined with any other element using the group operation, leaves that element unchanged. It is commonly denoted by e or 1.

What is an abelian group?

An abelian group is one in which the binary operation is commutative. This means for any two elements a and b in the group, a · b = b · a

A subgroup is a subset of a group that is itself a group under the same binary operation. It must satisfy the group properties: closure, associativity, identity, and inverses.

What is a normal subgroup and how does it relate to quotient groups?

A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group. This means for every element n in the normal subgroup N and every element g in the group, gng⁻¹ is still in N. Normal subgroups allow the construction of quotient groups, where the group is partitioned into cosets of the normal subgroup.

What are group homomorphisms?

A group homomorphism is a function between two groups that preserves the group operation. This means if f: G → H is a homomorphism and a, b are elements of G, then f(a · b) = f(a) · f(b) in H.

What is Lagrange’s theorem in group theory?

Lagrange's theorem states that for any finite group, the order (number of elements) of every subgroup divides the order of the entire group. This theorem is a fundamental result in the study of finite groups.

What is Cayley’s theorem?

Cayley’s theorem states that every group is isomorphic to a subgroup of a symmetric group. This implies that every group can be represented as a group of permutations acting on a set.

How is group theory applied in other fields?

Group theory has applications in many fields including: Physics: Describing symmetries and conservation laws. Chemistry: Analyzing molecular symmetry and chemical bonding. Cryptography: Underlying structures in cryptographic systems. Mathematics: Foundational in algebra, geometry, and number theory.

Home > Docs > Group Theory > Automorphism > Automorphism Group

Automorphism Group: Proof of Composition Structure

Bivash Majumder

Table of Contents

1.Introduction to Automorphism Group Structure
2.Theorem
3.Proof
3.1.1. Closure
3.2.2. Associativity
3.3.3. Identity Element
3.4.4. Inverses
3.5.Conclusion
3.6.Related Docs
4.Automorphism
4.1.FAQs
5.
6.Group Theory
7.Recent Posts on Study Materials
8.Recent Posts on Questions

Introduction to Automorphism Group Structure

In the captivating world of abstract algebra, the concept of an automorphism group offers a profound glimpse into the symmetry and internal consistency of algebraic systems. An automorphism is a bijective mapping that preserves the operation in a group $(G,\cdot)$ , ensuring that every element’s structure is maintained under the transformation.

This article delves into the elegant proof that the complete set of automorphisms, denoted as $\text{Aut}(G)$ , itself forms a group when the operation of function composition is applied. This result is not only a testament to the inherent order within mathematical structures but also a pivotal theorem that enriches our understanding of group theory.

Theorem

Prove that the set of all automorphisms of a group $(G,\cdot)$ forms a group with respect to the composition of mappings.

Proof

Let $\operatorname{Aut}(G)$ denote the set of all automorphisms of the group $(G, \cdot)$ . We show that $\operatorname{Aut}(G)$ forms a group under the composition of mappings by verifying the group axioms.

1. Closure

Let $f, g \in \operatorname{Aut}(G)$ . Since each is an isomorphism, both are homomorphisms and bijective. Consider the composition $f \circ g$ . For any $a, b \in G$ :

\begin{aligned} (f \circ g) (a \cdot b) & = f (g (a \cdot b)) \\ = f (g (a) \cdot g (b)) (since g is a homomorphism) \\ = f (g (a)) \cdot f (g (b)) (since f is a homomorphism) \\ = (f \circ g) (a) \cdot (f \circ g) (b) \end{aligned}

Thus, $f \circ g$ is a homomorphism. Being a composition of bijections, it is also bijective. Hence, $f \circ g$ is an automorphism, showing closure.

2. Associativity

For any $f, g, h \in \operatorname{Aut}(G)$ , function composition is inherently associative:

(f \circ g) \circ h = f \circ (g \circ h)

3. Identity Element

The identity mapping $\operatorname{I}_G : G \to G$ defined by $\operatorname{I}_G(a) = a$ for all $a \in G$ is an automorphism. It satisfies:

\begin{aligned} I_{G} (a \cdot b) & = a \cdot b \\ = I_{G} (a) \cdot I_{G} (b) \end{aligned}

Therefore, $\operatorname{I}_G$ acts as the identity element in $\operatorname{Aut}(G)$ .

4. Inverses

Let $f \in \operatorname{Aut}(G)$ . Since $f$ is a bijective homomorphism, its inverse $f^{-1}$ exists. For all $a, b \in G$ , using the fact that $f$ preserves the group operation, we have:

f^{-1}(a \cdot b) = f^{-1}(a) \cdot f^{-1}(b)

Hence, $f^{-1}$ is also an automorphism, providing the inverse for every element in $\operatorname{Aut}(G)$ .

Conclusion

Since the set $\operatorname{Aut}(G)$ is closed under composition, the operation is associative, an identity element exists, and every automorphism has an inverse, we conclude that $\operatorname{Aut}(G)$ forms a group under the composition of mappings.

Automorphism

FAQs

Group Theory

What is a group in group theory?

A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for every element.
What are the main properties of a group?
The four main properties are:
- Closure: The result of the operation on any two elements of the group is also in the group.
- Associativity: The group operation is associative.
- Identity: There exists an element that does not change other elements when used in the operation.
- Invertibility: Every element has an inverse that, when combined with the element, yields the identity.
What is the identity element in a group?

The identity element is a unique element in the group that, when combined with any other element using the group operation, leaves that element unchanged. It is commonly denoted by e or 1.
What is an abelian group?

An abelian group is one in which the binary operation is commutative. This means for any two elements a and b in the group, a · b = b · a
What is a subgroup?

A subgroup is a subset of a group that is itself a group under the same binary operation. It must satisfy the group properties: closure, associativity, identity, and inverses.
What is a normal subgroup and how does it relate to quotient groups?

A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group. This means for every element n in the normal subgroup N and every element g in the group, gng⁻¹ is still in N. Normal subgroups allow the construction of quotient groups, where the group is partitioned into cosets of the normal subgroup.
What are group homomorphisms?

A group homomorphism is a function between two groups that preserves the group operation. This means if f: G → H is a homomorphism and a, b are elements of G, then f(a · b) = f(a) · f(b) in H.
What is Lagrange’s theorem in group theory?

Lagrange's theorem states that for any finite group, the order (number of elements) of every subgroup divides the order of the entire group. This theorem is a fundamental result in the study of finite groups.
What is Cayley’s theorem?

Cayley’s theorem states that every group is isomorphic to a subgroup of a symmetric group. This implies that every group can be represented as a group of permutations acting on a set.
How is group theory applied in other fields?
Group theory has applications in many fields including:
- Physics: Describing symmetries and conservation laws.
- Chemistry: Analyzing molecular symmetry and chemical bonding.
- Cryptography: Underlying structures in cryptographic systems.
- Mathematics: Foundational in algebra, geometry, and number theory.

Knowledge Bases

Previous Year Question Papers

Vidyasagar University- Quesitions Papers

Group Theory

Automorphism Group: Proof of Composition Structure

Introduction to Automorphism Group Structure

Theorem

Proof

1. Closure

2. Associativity

3. Identity Element

4. Inverses

Conclusion

Related Docs

Automorphism

FAQs

Group Theory

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