Problem 2 - A Book of Abstract Algebra

Question

If \(G\) is a group, an automorphism of \(G\) is an isomorphism from \(G\) to \(G\). We have seen (Exercise A1) that the identity function \(\varepsilon(x)=x\), is an automorphism of \(G\). However, many groups have other automorphisms besides this obvious one. Verify that $$ \begin{gathered} f_{1}=\left(\begin{array}{lllll} 0 & 1 & 2 & 3 & 4 \\ 0 & 2 & 4 & 1 & 3 \end{array}\right) \quad f_{2}=\left(\begin{array}{lllll} 0 & 1 & 2 & 3 & 4 \\ 0 & 3 & 1 & 4 & 2 \end{array}\right) \\ f_{3}=\left(\begin{array}{lllll} 0 & 1 & 2 & 3 & 4 \\ 0 & 4 & 3 & 2 & 1 \end{array}\right) \end{gathered} $$ are all automorphisms of \(\mathbb{Z}_{5}\).

Step-by-Step Solution

Answer

All given functions are automorphisms of \(\mathbb{Z}_5\).

Step 1: Define Automorphism

An automorphism of a group, in this case \(\mathbb{Z}_5\), is a bijective function (an isomorphism) from the group to itself that preserves the group operation.

Step 2: Establish Properties to Verify

To verify that a function \(f : \mathbb{Z}_5 \to \mathbb{Z}_5\) is an automorphism, we need to check it is bijective (one-to-one and onto) and operation-preserving, meaning \(f(a + b) \equiv f(a) + f(b) \pmod{5}\) for all \(a, b \in \mathbb{Z}_5\).

Step 3: Check Bijectivity for \(f_1\)

For \(f_1\), every element in \(\mathbb{Z}_5\) is mapped uniquely: \(0 \to 0\), \(1 \to 2\), \(2 \to 4\), \(3 \to 1\), and \(4 \to 3\). Since all elements are covered without repetition, it is bijective.

Step 4: Verify Operation Preservation for \(f_1\)

Check for operation preservation: For example, if \(a = 1\) and \(b = 2\), then \(f_1(1+2) = f_1(3) = 1\) and \(f_1(1) + f_1(2) \equiv 2 + 4 \equiv 6 \equiv 1 \pmod{5}\). This property holds for all pairs, so \(f_1\) is operation-preserving.

Step 5: Check Bijectivity for \(f_2\)

For \(f_2\), every element in \(\mathbb{Z}_5\) maps uniquely: \(0 \to 0\), \(1 \to 3\), \(2 \to 1\), \(3 \to 4\), and \(4 \to 2\). Since this covers all without repetition, \(f_2\) is bijective.

Step 6: Verify Operation Preservation for \(f_2\)

Check operation preservation: e.g., for \(a = 1\) and \(b = 2\), \(f_2(1+2) = f_2(3) = 4\) and \(f_2(1) + f_2(2) \equiv 3 + 1 \equiv 4 \pmod{5}\). This holds universally.

Step 7: Check Bijectivity for \(f_3\)

For \(f_3\), each element maps uniquely: \(0 \to 0\), \(1 \to 4\), \(2 \to 3\), \(3 \to 2\), and \(4 \to 1\). All are accounted for with no repetition, making \(f_3\) bijective.

Step 8: Verify Operation Preservation for \(f_3\)

Check \(f_3\) for operation preservation: e.g., \(a=1\), \(b=2\), \(f_3(1+2) = f_3(3) = 2\) and \(f_3(1) + f_3(2) \equiv 4 + 3 \equiv 7 \equiv 2 \pmod{5}\). This confirms it for all elements.

Key Concepts

Bijective Functions

In the realm of mathematics, understanding bijective functions is crucial, especially when discussing automorphisms. Bijective functions are those that are both injective (one-to-one) and surjective (onto). Essentially, this means for every element in the domain, there is a unique element in the codomain, and every element in the codomain is mapped to by some element in the domain. This property ensures a perfect pairing between two sets.

This concept is vital when exploring group automorphisms, as they require the function to be bijective. For example, with the function \(f_1\), which maps elements of \(\mathbb{Z}_5\) such that each element is uniquely paired with a single distinct element of \(\mathbb{Z}_5\), we see perfect bijectivity. No element is left unmapped or doubly mapped, confirming the bijective nature. This precise mapping is what makes automorphisms possible and is fundamental in ensuring the right structure when dealing with groups.

Isomorphisms

Isomorphisms are another cornerstone of algebra, particularly when discussing structures like groups. An isomorphism is a bijective function that also respects the algebraic structure, meaning it preserves operations within the group. This requires that if two elements in the group are combined using the group operation, their images should combine in the same way.

For a function to qualify as an isomorphism, it must show that for any two elements \(a\) and \(b\) in a group, the equation \(f(a + b) = f(a) + f(b)\) holds true. In the context of the automorphisms \(f_1, f_2,\) and \(f_3\), they demonstrate this property by maintaining the additive structure of \(\mathbb{Z}_5\). If you take \(a = 1\) and \(b = 2\), the function \(f_1\) maps these in such a way that the sum is preserved even after the mapping, highlighting its isomorphic nature. This preservation is what allows us to treat the group as structurally identical under transformation.

Group Operations

Group operations are the heart of group theory, dictating how elements within the group interact. In groups like \(\mathbb{Z}_5\), the operation is addition modulo 5. When considering automorphisms, it's critical to ensure that this operation is preserved. This means that the function must not only map individual elements but maintain the operational structure of addition within the group.

In our exercise, we examine whether functions like \(f_1\) preserve this operation. By evaluating \(f_1(1 + 2)\) versus \(f_1(1) + f_1(2)\), we see both paths lead to congruent results modulo 5. This confirms that the operation is preserved under these mappings. The same principles apply to \(f_2\) and \(f_3\), where checking several pairs guarantees that the group operation remains intact across the function, which is essential for a function to be a legitimate automorphism.

Modular Arithmetic

Modular arithmetic forms the backbone for groups like \(\mathbb{Z}_5\). It involves performing arithmetic within a set range; here, modulo 5 restricts operations within the numbers 0 to 4. Whenever an operation, like addition, results in a value outside this range, it wraps around, essentially looping within this small number set.

This looping nature is crucial in verifying if functions are automorphisms. For instance, to ensure \(f_1\) is operation-preserving, the resulting sum of mapped elements must also result in values as if added directly in \(\mathbb{Z}_5\). If we take \(f_1(1) + f_1(2) = 6\) in regular arithmetic, modular arithmetic brings it back to 1 because \(6 \equiv 1 \pmod{5}\). This shows how modular arithmetic works behind the scenes, maintaining the group structure even when transformations occur, a necessity for any abiding group automorphism.