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 > Trivial Inner Automorphisms And Commutativity In Groups

Trivial Inner Automorphisms and Commutativity in Groups

Bivash Majumder

Table of Contents

Introduction

Welcome to an enlightening exploration of a cornerstone theorem in group theory that bridges the concept of inner automorphisms with the nature of commutativity. In any group \( G \), the theorem states that \(|\operatorname{Inn}(G)| = 1\) if and only if \(G\) is abelian, meaning every element commutes with every other. This pivotal result reveals that a group with only trivial inner automorphisms exhibits a harmonious structure where conjugation acts as the identity.

As you delve into this fascinating topic, you’ll gain a deeper understanding of how trivial inner automorphisms not only simplify the intricate symmetries within groups but also serve as a definitive marker of abelian groups. We encourage you to immerse yourself in the proof, explore further examples, and challenge your mathematical intuition. Take the next step in your journey through abstract algebra and unlock the elegant secrets of group theory!

Theorem

Let \(G\) be a group. Prove that \(|\operatorname{Inn}(G)| = 1\) if and only if \(G\) is commutative.

Proof

Step 1

Let us assume \(|\operatorname{Inn}(G)| = 1\)

Since \(|\operatorname{Inn}(G)| = 1\), the only inner automorphism of \(G\) is the identity map, denoted by \(\operatorname{I_{G}}\). Thus, for every \(g \in G\), we have:

\[ \theta_g(x) = gxg^{-1} = x \quad \text{for all } x \in G \nonumber \]

Implies that:

\[ gx = xg \quad \text{for all } g,x \in G\nonumber \]

Hence, \(G\) is commutative (abelian).

Step 2

Let us assume \(G\) is commutative

If \(G\) is abelian, then for every \(g, x \in G\) we have:

\[ \theta_g(x) = gxg^{-1} = xgg^{-1} = x\nonumber \]

Therefore, each inner automorphism \(\theta_g\) is the identity map \(\operatorname{I_{G}}\). This shows that:

\[ \operatorname{Inn}(G) = \{\operatorname{I_{G}}\}\nonumber \]

and consequently, \(|\operatorname{Inn}(G)| = 1\).

Summary

We have shown that if \(|\operatorname{Inn}(G)| = 1\), then every inner automorphism is trivial, implying \(gxg^{-1} = x\) for all \(g, x \in G\), which means \(G\) is commutative. Conversely, if \(G\) is abelian, every inner automorphism is the identity map, so \(|\operatorname{Inn}(G)| = 1\). This completes the proof.

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

Trivial Inner Automorphisms and Commutativity in Groups

Introduction

Theorem

Proof

Step 1

Step 2

Summary

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Automorphism

FAQs

Group Theory

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