Table of Contents

    Introduction

    Group theory, a cornerstone of modern mathematics, reveals the captivating interplay between structure and symmetry. In this exploration, we spotlight the concept of inner automorphisms—transformations that preserve the essential properties of a group—and explain how they are isomorphic to the quotient group obtained by dividing the group \(G\) by its center \(Z(G)\). This foundational isomorphism, expressed as \( \operatorname{Inn}(G) \cong G/Z(G) \), unveils deep insights into the behavior of groups and their inherent symmetries. Prepare to immerse yourself in a clear, engaging journey through definitions, theorems, and compelling proofs that illuminate one of the most elegant results in group theory.

    Theorem


    Prove that \(\operatorname{Inn}(G) \cong G/Z(G)\), where \(\operatorname{Inn}(G)\) is the group of inner automorphisms of \(G\) and \(Z(G)\) is the centre of \(G\).


    Proof

    Step 1

    Define the Mapping

    Define \(\phi: G \to \operatorname{Inn}(G)\) by \[ \phi(g) = \theta_g, \quad \text{where } \theta_g(x) = g x g^{-1} \text{ for all } x \in G. \]

    Step 2

    To show that \(\phi\) is a Homomorphism

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

    \[ \begin{align} \phi(gh)(x) &= \theta_{gh}(x) = (gh)x(gh)^{-1} \nonumber\\ &= g\,(h x h^{-1})\,g^{-1} \nonumber\\ &= \theta_g\big(\theta_h(x)\big) \nonumber\\ &= (\theta_g \circ \theta_h)(x) \nonumber \end{align} \]

    Thus, \(\phi(gh) = \theta_g \circ \theta_h = \phi(g) \phi(h)\), showing that \(\phi\) is a group homomorphism.

    Step 3

    To determine the Kernel of \(\phi\)

    The kernel of \(\phi\) is given by:

    \[ \ker(\phi) = \{g \in G \mid \theta_g(x) = x \text{ for all } x \in G\}. \]

    Since \(\theta_g(x) = g x g^{-1} = x\) for all \(x \in G\) if and only if \(gx = xg\) for all \(x \in G\), we have:

    \[ \ker(\phi) = Z(G). \]

    Step 4

    Apply the First Isomorphism Theorem

    The First Isomorphism Theorem states that:

    \[ G/\ker(\phi) \cong \operatorname{Im}(\phi). \]

    Since \(\operatorname{Im}(\phi) = \operatorname{Inn}(G)\) and \(\ker(\phi) = Z(G)\), it follows that:

    \[ G/Z(G) \cong \operatorname{Inn}(G). \]

    Summary


    We defined the homomorphism \(\phi: G \to \operatorname{Inn}(G)\) by \(\phi(g) = \theta_g\) and verified that it is a group homomorphism. Its kernel is the centre \(Z(G)\), and by the First Isomorphism Theorem, we conclude that \(G/Z(G) \cong \operatorname{Inn}(G)\).

    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.
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