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    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.

    Theorem


    Let \( (G,\cdot) \) be a group and \( H \) be a subgroup of \( G \). Then \( \text{Aut}(H) \) is a subgroup of \( \text{Aut}(G) \).

    Proof

    Given that \( (G,\cdot) \) is a group and \( H \) is a subgroup of \( G \).
    To prove \( \text{Aut}(H) \) is a subgroup of \( \text{Aut}(G) \).

    Step 1

    To prove \( \text{Aut}(H) \subseteq \text{Aut}(G) \)

    Let \( \alpha \in \text{Aut}(H) \) \( \implies \alpha:H \to H \) is an isomorphism. Then \( \alpha:H \to G \) is a monomorphism since \( H \subseteq G \). Let \( \beta:G \to G \) be a mapping such that \[ \beta(x) = \begin{cases} \alpha(x) & \text{when } x \in H \\ x & \text{when } x \in G – H \end{cases} \] \( \implies \beta|_{H} = \alpha \). That is, \( \alpha:H \to G \) is a restriction of \( \beta:G \to G \).
    Clearly, \( \beta \) is bijective and a homomorphism.
    \( \implies \beta \) is an isomorphism.
    \( \implies \alpha \in \text{Aut}(G) \).
    Therefore, \( \text{Aut}(H) \subseteq \text{Aut}(G) \).

    Step 2

    To prove \( \text{Aut}(H) \neq \Phi \)

    Since \( I_{H} \) is the identity mapping on \( H \), and it is also bijective and a homomorphism, Therefore, \( I_{H} \in \text{Aut}(H) \).
    \( \implies \text{Aut}(H) \neq \Phi \).

    Step 3

    To prove \( \alpha \circ \beta^{-1} \in \text{Aut}(H) ~\forall~ \alpha, \beta \in \text{Aut}(H) \)

    Let \( \alpha, \beta \in \text{Aut}(H) \)
    \( \implies \alpha, \beta \) are isomorphisms.
    \( \implies \alpha, \beta^{-1} \) are isomorphisms.
    Since \( \alpha:H \to H \) and \( \beta^{-1}:H \to H \), therefore \( \alpha \circ \beta^{-1}:H \to H \) exists.

    Step 4

    To prove \( \alpha \circ \beta^{-1} \) is bijective.

    Since \( \alpha \) and \( \beta^{-1} \) are bijective, then \( \alpha \circ \beta^{-1} \) is bijective.

    Step 5

    To prove \( \alpha \circ \beta^{-1} \) is a homomorphism.

    Let \( x, y \in H \). \[ \begin{aligned} (\alpha \circ \beta^{-1})(x \cdot y) &= \alpha(\beta^{-1}(x \cdot y)) \\ &= \alpha(\beta^{-1}(x) \cdot \beta^{-1}(y)) \\ &= \alpha(\beta^{-1}(x)) \cdot \alpha(\beta^{-1}(y)) \\ &= (\alpha \circ \beta^{-1})(x) \cdot (\alpha \circ \beta^{-1})(y) \end{aligned} \] Therefore, \( \alpha \circ \beta^{-1} \) is a homomorphism.

    Conclusion

    Therefore, \( \alpha \circ \beta^{-1} \in \text{Aut}(H) \).
    Hence, \( \text{Aut}(H) \) is a subgroup of \( \text{Aut}(G) \).

    Conclusion


    In conclusion, mastering the concept of subgroup automorphism enriches your appreciation for group theory by revealing the layered and harmonious structure of automorphism groups. Embark on this mathematical journey and let the profound insights of subgroup automorphism inspire your continued exploration of algebra!

    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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