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Group Automorphism: Definition and Inverse Mapping

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Introduction to Group Automorphism

Welcome to our in‐depth exploration of group automorphism—a cornerstone concept in modern abstract algebra. In this engaging overview, we unravel how an automorphism, an isomorphism that maps a group onto itself while preserving its structure, offers profound insights into the symmetry and intrinsic properties of algebraic systems.

Definition: Automorphism

An automorphism of a group GG is an isomorphism from GG onto itself. In other words, a map φ:G→G\varphi: G \to G is an automorphism if it is a homomorphism, bijective, and its inverse is also a homomorphism.

Theorem

Let (G,⋅)(G,\cdot) be a group and let α:G→G\alpha: G \to G be defined by α(x)=x−1\alpha(x) = x^{-1} for all x∈Gx \in G. Then α\alpha is an automorphism if and only if GG is abelian.

Proof

Step 1

Let α\alpha is an Automorphism

Since α\alpha is an automorphism, it is a homomorphism. Therefore, for all x,y∈Gx, y \in G,

α(xy)=α(x)α(y)α(xy)=α(x)α(y)\begin{align} \alpha(xy) &= \alpha(x) \alpha(y) \tag{1} \end{align}

By definition, α(x)=x−1\alpha(x) = x^{-1}. Thus, equation (1) becomes:

(xy)−1=x−1y−1.(xy)−1=x−1y−1.\begin{align} (xy)^{-1} &= x^{-1}y^{-1}. \tag{2} \end{align}

However, from the general property of groups, we have:

(xy)−1=y−1x−1.(xy)−1=y−1x−1.\begin{align} (xy)^{-1} &= y^{-1}x^{-1}. \tag{3} \end{align}

Equating (2) and (3) gives:

x−1y−1=y−1x−1.x−1y−1=y−1x−1.\begin{align} x^{-1}y^{-1} &= y^{-1}x^{-1}. \tag{4} \end{align}

Taking inverses on both sides (noting that the inverse of an inverse returns the original element), we obtain:

xy=yx.xy=yx.\begin{align} xy &= yx. \tag{5} \end{align}

Hence, GG is abelian.

Step 2

Let GG is Abelian

Conversely, suppose GG is abelian. Then for all x,y∈Gx, y \in G, we have:

α(xy)=(xy)−1=x−1y−1(since xy=yx)=α(x)α(y).α(xy)=(xy)−1=x−1y−1(since xy=yx)=α(x)α(y).\begin{align} \alpha(xy) &= (xy)^{-1} \nonumber\\[5mm] &= x^{-1}y^{-1} \quad \text{(since \(xy = yx\))} \nonumber\\[5mm] &= \alpha(x)\alpha(y). \tag{6} \end{align}

Thus, α\alpha is a homomorphism. Also, since inversion is a bijective mapping on any group (with α\alpha being its own inverse), α\alpha is an automorphism.

Summary

We have shown that if the inversion map α(x)=x−1\alpha(x)=x^{-1} is an automorphism, then it forces the relation x−1y−1=y−1x−1x^{-1}y^{-1} = y^{-1}x^{-1}, which implies that xy=yxxy = yx for all x,y∈Gx,y \in G, meaning GG is abelian. Conversely, if GG is abelian, then the inversion map is a homomorphism and hence an automorphism. Therefore, α\alpha is an automorphism if and only if GG is abelian.

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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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