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 > Inner Automorphism Composition In Group Theory

Inner Automorphism Composition in Group Theory

Bivash Majumder

Table of Contents

Introduction to Inner Automorphism Composition

Embark on an enlightening journey into the heart of group theory, where the concept of inner automorphism unveils the intrinsic symmetries that govern algebraic structures. At the core of this exploration is a fascinating theorem: for any group \(G\) and elements \(a, b \in G\), the composition of inner automorphisms, represented by \[ \theta_a \circ \theta_b = \theta_{ab}, \] perfectly encapsulates how these transformations preserve the group’s structure.

This theorem not only highlights the elegant interplay between group elements through conjugation but also emphasizes the consistent order that underpins abstract algebra. Whether you are a seasoned mathematician or a curious beginner, understanding inner automorphism composition offers profound insights into the symmetry and balance of mathematical systems.

Theorem

Let \(G\) be a group and let \(a,b\in G\). Prove that the composition of the inner automorphisms corresponding to \(a\) and \(b\) satisfies \[ \theta_a \circ \theta_b =\theta_{ab}, \] where \(\theta_a\) and \(\theta_b\) denote the inner automorphisms corresponding to \(a\) and \(b\), respectively.

Proof

Step 1

Define the Inner Automorphisms

For any element \(g \in G\), the inner automorphism \(\theta_g\) is defined by:

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

Step 2

To compute the Composition \(\theta_a \circ \theta_b\)

For any \(x \in G\), we have:

\[ \begin{align} (\theta_a \circ \theta_b)(x) &= \theta_a\big(\theta_b(x)\big) \nonumber\\ &= \theta_a\big(b x b^{-1}\big) \nonumber\\ &= a\,(b x b^{-1})\, a^{-1} \nonumber\\ &= (ab)\, x\, (b^{-1}a^{-1}) \nonumber\\ &= (ab)\, x\, (ab)^{-1} \end{align} \]

Step 3

Conclusion

Equation (1) shows that for all \(x \in G\):

\[ (\theta_a \circ \theta_b)(x) = \theta_{ab}(x). \]

Hence, we conclude that: \[ \theta_a \circ \theta_b = \theta_{ab}. \]

Summary

We defined the inner automorphisms \(\theta_a\) and \(\theta_b\) by \(\theta_a(x) = axa^{-1}\) and \(\theta_b(x) = bxb^{-1}\). By composing these maps, we showed that: \[ (\theta_a \circ \theta_b)(x) = (ab)x(ab)^{-1}, \] which means \(\theta_a \circ \theta_b = \theta_{ab}\) for all \(x \in G\). 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

Inner Automorphism Composition in Group Theory

Introduction to Inner Automorphism Composition

Theorem

Proof

Step 1

Step 2

Step 3

Summary

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Automorphism

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Group Theory

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