Problem 1 - A Book of Abstract Algebra

Question

The following three facts about isomorphism are true for all groups: 1\. Every group is isomorphic to itself. 2\. If \(G_{1} \cong G_{2}\), then \(G_{2} \cong G_{1}\). 3\. If \(G_{1} \cong G_{2}\) and \(G_{2} \cong G_{3}\), then \(G_{1} \cong G_{3}\). Fact 1 asserts that for any group \(G\), there exists an isomorphism from \(G\) to \(G\). Fact 2 asserts that, if there is an isomorphism \(f\) from \(G_{1}\) to \(G_{2}\), there must be some isomorphism from \(G_{2}\) to \(G_{1}\). Well, the inverse of \(f\) is such an isomorphism. Fact 3 asserts that, if there are isomorphisms \(f: G_{1} \rightarrow G_{2}\) and \(g: G_{2} \rightarrow G_{3}\), there must be an isomorphism from \(G_{1}\) to \(G_{3}\). One can easily guess that \(g \circ f\) is such an isomorphism. The details of facts 1,2 , and 3 are left as exercises. Let \(G\) be any group. If \(\varepsilon: G \rightarrow G\) is the identity function, \(\varepsilon(x)=x\), show that \(\varepsilon\) is an isomorphism.

Step-by-Step Solution

Answer

The identity function \( \varepsilon: G \rightarrow G \) is an isomorphism because it is bijective and preserves group operations.

Step 1: Define the Identity Function

The identity function \( \varepsilon: G \rightarrow G \) maps each element of the group \( G \) to itself. Therefore, for any \( x \in G \), we have \( \varepsilon(x) = x \). This function is often called the identity function due to this characteristic.

Step 2: Prove Bijectiveness of \(\varepsilon\)

A function is an isomorphism if it is a bijective homomorphism. We start by proving bijectiveness. For the identity function \( \varepsilon \), each element maps uniquely to itself. Thus, \( \varepsilon \) is injective (one-to-one) because no two different elements can map to the same element. Also, \( \varepsilon \) is surjective (onto) because every element in \( G \) has a preimage in \( G \). Thus, \( \varepsilon \) is bijective.

Step 3: Prove Homomorphic Property of \(\varepsilon\)

To be a homomorphism, \( \varepsilon \) must preserve group operations. If \( \cdot \) is the group operation in \( G \), for any \( a, b \in G \), \( \varepsilon(a\cdot b) = a\cdot b = \varepsilon(a) \cdot \varepsilon(b) \). Therefore, \( \varepsilon \) preserves the operation of the group.

Step 4: Conclusion

Since \( \varepsilon \) is bijective and a homomorphism, it is an isomorphism between \( G \) and itself. Therefore, the identity function \( \varepsilon: G \rightarrow G \) is an isomorphism.

Key Concepts

Group Theory

Group theory is a significant branch of mathematics focusing on the study of algebraic structures known as groups. A group is a set of elements combined with an operation that satisfies four primary conditions:

• Closure: For every pair of elements in the group, the result of the operation on these two elements is also in the group.
• Associativity: The operation is associative; meaning for any three elements in the group, the equation \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) holds.


• Identity Element: There is an identity element in the group, such that for any element \( a \), the equation \( a \cdot e = e \cdot a = a \) holds.
• Inverse Element: For every element in the group, there is an inverse element such that when the operation is applied between these two, the identity element results.

Understanding these foundational properties allows us to explore deeper concepts such as group isomorphism, which addresses the idea of structural similarity between groups. When two groups are isomorphic, they have the same structure, implying that there's a one-to-one correspondence between the elements that also preserves operation.

Identity Function

The identity function is a crucial concept when discussing group isomorphisms. Defined as \( \varepsilon: G \to G \), this function maps every element of a group \( G \) to itself. This seems simple, but it's fundamental in proving concepts like a group's isomorphism to itself.
To visualize this, consider a group where every element \( x \) gets mapped back exactly to \( x \). This function is a perfect example of bijectiveness, as each element uniquely leads to itself, and no element is left without a preimage. Hence, it's both injective (no two elements map to the same element) and surjective (every element is mapped).
The identity function retains group operations. Given any elements \( a, b \) in the group, the identity function ensures that \( \varepsilon(a \cdot b) = a \cdot b = \varepsilon(a) \cdot \varepsilon(b) \). This confirms that the identity function preserves how operations work within the group itself.

Bijective Homomorphism

A bijective homomorphism is a mapping between groups that is both a homomorphism and bijective. To understand this, consider two essential qualities:

• Homomorphism: This property means the function must preserve group operations. That is, if \( f: G \to H \) is a homomorphism, then for any elements \( a, b \in G \), it satisfies \( f(a \cdot b) = f(a) \cdot f(b) \).


• Bijectiveness: The function must be one-to-one (injective) and onto (surjective). This ensures that every element in the first group maps uniquely to an element in the second group, covering all elements.

Let's consider an identity function \( \varepsilon \), which maps every element of Group \( G \) back to itself. It beautifully exemplifies a bijective homomorphism. Each element has an exact counterpart (itself), validating both injective and surjective properties. It also maintains group operations, reinforcing it as a true homomorphic function.

Group Operations

Group operations form the backbone of group theory. They define the way elements combine within a group. To be a valid group operation, it must satisfy properties like closure, associativity, and be performed with respect to an identity element.
Let's explore these operations further:

• Closure: Whenever you combine two elements from the group using the group operation, the result must still be an element of the group.


• Associativity: Group operations are required to be associative. This means that changing the grouping of the elements does not change the result of the operation.
• Identity: Every group has an identity element. This special element, when combined with any element of the group, leaves it unchanged.
• Inverses: Every element in the group must have a corresponding inverse element such that their combination yields the identity element.

Understanding these operations is essential for working with groups and exploring deeper concepts like group isomorphism, which relies on preserving these operations between groups.