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    Introduction to Inner Automorphism and Normal Subgroups

    In the fascinating realm of abstract algebra, the concept of inner automorphism serves as a gateway to understanding the inherent symmetry and structure of groups. Through the process of conjugation, defined by \[ \theta_{a}(x)=a\cdot x\cdot a^{-1}\quad \text{for all } x\in G, \] inner automorphisms elegantly reconfigure group elements while preserving the fundamental operation.

    The theorem under discussion establishes that for any group \((G,\cdot)\), the collection of inner automorphisms, denoted by \(\operatorname{Inn}(G)\), forms a normal subgroup of the full automorphism group \(\operatorname{Aut}(G)\). This pivotal result not only underscores the beauty of group symmetries but also provides a solid foundation for advanced studies in group theory.

    Introduction to Subgroup Automorphism in Group Theory Welcome to an insightful exploration into subgroup automorphism within the fascinating world of group theory. In this discussion, we examine a fundamental theorem: if \( (G,\cdot) \) is a group and \( H \) is a subgroup of \( G \), then the automorphism set of \( H \), denoted by \( \text{Aut}(H) \), constitutes a subgroup of the entire automorphism group \( \text{Aut}(G) \). This elegant result not only demonstrates the inherent symmetry in algebraic structures but also emphasizes the deep interconnections that exist between groups and their automorphic mappings.

    Definition: Inner Automorphism


    Let \( (G,\cdot) \) be a group. The set of all automorphisms \( \theta_{a}:G\to G \) for all \( a\in G \), defined by \[ \theta_{a}(x)=a\cdot x\cdot a^{-1} \quad \forall~x\in G, \] is said to be the Inner Automorphism of \( G \) and is denoted by \( \text{Inn}(G) \). i.e., \[ \text{Inn}(G)=\{\theta_{a} \mid \theta_{a}:G \to G \text{ defined by } \theta_{a}(x)=a\cdot x\cdot a^{-1} \quad \forall~x\in G \text{ and } a\in G \}. \]

    Theorem


    For any group \((G,\cdot)\), prove that \(\operatorname{Inn}(G)\) is a normal subgroup of \(\operatorname{Aut}(G)\).


    Proof

    Step 1:

    To Prove \(\operatorname{Inn}(G)\) is a Subgroup of \(\operatorname{Aut}(G)\)

    For each \(g \in G\), define the mapping \[ \theta_g : G \to G \quad \text{by} \quad \theta_g(x) = g x g^{-1}. \] The set \[ \operatorname{Inn}(G) = \{ \theta_g \mid g \in G \} \]

    Closure: Let \(\theta_g, \theta_h \in \operatorname{Inn}(G)\). Then for any \(x \in G\):

    \[ \begin{align} (\theta_g \circ \theta_h)(x) &= \theta_g\big(\theta_h(x)\big) \nonumber\\[5mm] &= \theta_g\big(h x h^{-1}\big) \nonumber\\ &= g \big(h x h^{-1}\big) g^{-1} \nonumber\\ &= (gh) \, x \, (gh)^{-1} \nonumber\\ &= \theta_{gh}(x)\nonumber \end{align} \]

    Since \(gh \in G\), the mapping \(\theta_{gh}\) is an element of \(\operatorname{Inn}(G)\). Thus, \(\operatorname{Inn}(G)\) is closed under composition.

    Identity: For the identity element \(e \in G\), \[ \theta_e(x) = e x e^{-1} = x, \] so \(\theta_e\) is the identity automorphism.

    Inverses: For any \(g \in G\), the inverse of \(\theta_g\) is given by \[ \theta_g^{-1} = \theta_{g^{-1}}, \] since for any \(x \in G\):

    \[ \theta_g \circ \theta_{g^{-1}}(x) = \theta_{gg^{-1}}(x) = \theta_e(x) = x. \]

    Therefore, \(\operatorname{Inn}(G)\) is a subgroup of \(\operatorname{Aut}(G)\).

    Step 2:

    To Prove \(\operatorname{Inn}(G)\) is Normal in \(\operatorname{Aut}(G)\)

    To show normality, we need to prove that for every \(\varphi \in \operatorname{Aut}(G)\) and every \(\theta_g \in \operatorname{Inn}(G)\), the conjugate \(\varphi \circ \theta_g \circ \varphi^{-1}\) is in \(\operatorname{Inn}(G)\).

    Let \(\varphi \in \operatorname{Aut}(G)\) and \(g \in G\). For any \(x \in G\):

    \[ \begin{align} (\varphi \circ \theta_g \circ \varphi^{-1})(x) &= \varphi\Big(\theta_g\big(\varphi^{-1}(x)\big)\Big) \nonumber\\[5mm] &= \varphi\Big(g\, \varphi^{-1}(x)\, g^{-1}\Big) \nonumber\\[5mm] &= \varphi(g) \, \varphi\big(\varphi^{-1}(x)\big) \, \varphi(g)^{-1} \nonumber\\[5mm] &= \varphi(g) \, x \, \varphi(g)^{-1} \end{align} \]

    Equation (1) shows that \[ \varphi \circ \theta_g \circ \varphi^{-1} = \theta_{\varphi(g)}, \] which is an element of \(\operatorname{Inn}(G)\). Thus, \(\operatorname{Inn}(G)\) is normal in \(\operatorname{Aut}(G)\).

    Summary


    We first established that \(\operatorname{Inn}(G)\) is a subgroup of \(\operatorname{Aut}(G)\) by verifying the identity, closure, and inverses. We then demonstrated that conjugation by any automorphism in \(\operatorname{Aut}(G)\) preserves the structure of inner automorphisms, confirming that \(\operatorname{Inn}(G)\) is a normal subgroup of \(\operatorname{Aut}(G)\).

    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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