Problem 40 - A First Course in Abstract Algebra

Question

Let \(\langle G,-\rangle\) be a group. Consider the binary operation * on the set \(G\) defined by $$ a+b=b \cdot a $$ for \(a, b \in G\). Show that \((G, \theta)\) is a group and that \((G, *)\) is actually isomorphte to \(\\{G, \cdot\\} .[\) Hint Consider the \(\operatorname{map} \phi\) with \(\phi(a)=a^{\prime}\) for \(a \in G .1\)

Step-by-Step Solution

Answer

\((G, *)\) is a group, and it's isomorphic to \(\langle G, \cdot \rangle\).

Step 1: Understand Group Properties

To show that \( (G, *) \) is a group, we need to verify the group properties: closure, associativity, identity, and existence of inverses under the operation \( * \).

Step 2: Verify Closure

Closure means that for all \( a, b \in G \), the result of \( a * b = b \cdot a \) is also in \( G \). In \( \langle G, \cdot \rangle \), \( b \cdot a \) is an element of \( G \) because the group is closed under its operation.

Step 3: Verify Associativity

For associativity, we need \( (a * b) * c = a * (b * c) \) for all \( a, b, c \in G \). This becomes \( (b \cdot a) \cdot c = c \cdot (b \cdot a) \) in the group \( \langle G, \cdot \rangle \), which holds due to associativity of \( \cdot \).

Step 4: Identity Element

The identity element \( e \) for \( * \) is such that for any \( a \in G \), \( e * a = a \) and \( a * e = a \). Using the same element from \( \langle G, \cdot \rangle \), we have \( a \cdot e = a \) and \( e \cdot a = a \), so \( e * a = a \) and \( a * e = a \) are satisfied.

Step 5: Existence of Inverses

For each element \( a \in G \), there exists an inverse \( a^{-1} \) such that \( a * a^{-1} = e \) and \( a^{-1} * a = e \). These are also true in \( \langle G, \cdot \rangle \), confirming the inverses under \( * \).

Step 6: Define the Isomorphism

To show that \( (G, *) \) is isomorphic to \( \langle G, \cdot \rangle \), consider the map \( \phi: G \rightarrow G \) where \( \phi(a) = a' \) such that \( a' = a \). This map must be bijective and homomorphic.

Step 7: Verify Bijectivity

The map \( \phi(a) = a \) is bijective because it is both injective (distinct inputs have distinct outputs) and surjective (every element in \( G \) has a pre-image).

Step 8: Verify Homomorphism

To check for homomorphism, ensure that \( \phi(a * b) = \phi(a) \cdot \phi(b) \). We have \( \phi(a * b) = \phi(b \cdot a) = b \cdot a \), and \( \phi(a) \cdot \phi(b) = a \cdot b \). Since these equal for \( * \) and \( \cdot \), the map preserves group operation.

Step 9: Conclusion of Isomorphism

As \( \phi \) is bijective and a homomorphism, \( (G, *) \) is isomorphic to \( \langle G, \cdot \rangle \) satisfying the problem's conditions.

Key Concepts

Group Properties

In the realm of group theory, understanding the properties that define a group is crucial. A group must satisfy four key properties: closure, associativity, identity, and the existence of inverses. These properties ensure a well-structured and predictable mathematical set under a given operation.

• Closure: This property ensures that if you take any two elements from a group and combine them using the group's operation, the result is also a member of the group. If we consider the group \( \langle G, \cdot \rangle \), any operation \( b \cdot a \) results in another element within \( G \).
• Associativity: This means the way in which elements are grouped in an operation doesn't affect the result. More formally, \((a * b) * c = a * (b * c)\). Applying this to our group operation \((G, *)\), it translates to \((b \cdot a) \cdot c = b \cdot (a \cdot c)\), which holds true since \( \cdot \) is associative.
• Identity Element: An element \( e \) exists such that for any element \( a \) in \( G \), both \( e \cdot a \) and \( a \cdot e \) return \( a \). This \( e \) is the identity element and works equally in our defined operation *.
• Existence of Inverses: For every element \( a \) in the group, there is an \( a^{-1} \) such that \( a \cdot a^{-1} = e \) and \( a^{-1} \cdot a = e \). In the new operation * on \( G \), every element still retains its inverse under the multiplication \( \cdot \).

Understanding these properties helps confirm whether a set with a specific operation qualifies as a group.

Isomorphism

In mathematics, two groups are considered isomorphic if there exists a bijective homomorphism between them. This means they share a structure that allows them to be transformed into one another without loss of their inherent properties. To demonstrate that the group \((G, *)\) is isomorphic to \(\langle G, \cdot \rangle \), we can use the map \(\phi\) defined for each element \( a \) as \( \phi(a) = a' \), with \( a'=a \).

• Bijectivity: A function is bijective if it is both injective and surjective. In this case, the map \( \phi(a) = a \) is straightforwardly bijective since each element maps uniquely to itself, ensuring distinct inputs produce distinct outputs (injective) and each element in \( G \) can be reached by some input (surjective).
• Homomorphism: A homomorphism is a function between two groups that respects the group operation. Specifically, this means \( \phi(a * b) = \phi(a) \cdot \phi(b) \). For the given operations, since \( \phi(a * b) = \phi(b \cdot a) = b \cdot a \) and \( \phi(a) \cdot \phi(b) = a \cdot b \), the operation is preserved, establishing the homomorphism.

Through both bijectivity and homomorphism, we establish that the groups are isomorphic, indicating their structural similarity.

Binary Operations

A binary operation is a rule for combining two elements of a set to produce another element of the same set. In our context, the operation \(*\) is defined on \(G\) such that for any elements \(a, b \in G\), \(a * b = b \cdot a\). This operation must fulfill specific properties to maintain the group's structure.

• Understanding the Operation: Here, \(a * b = b \cdot a\). Notice the change in order; it's essentially the regular multiplicative operation in the group \(\langle G, \cdot \rangle\) but flipped. Understanding how this reversal impacts operation is pivotal for manipulating the group's elements.
• Associating with Group Properties: Our operation \(*\) is shown to fulfill all critical group properties when analyzed under conclusions about closure, associativity, identity, and inverses already known in \(\langle G, \cdot \rangle\). This ensures \((G, *)\) is indeed a group.
• Importance of Binary Operations: Binary operations define the behavior and structure of groups, fundamentally shaping how we perform and interpret operations with group elements. In this problem, defining an unusual operation \(*\) while maintaining group characteristics offers insights into the flexibility and creativity in group theory.

The nature of binary operations like \(*\) helps illustrate the structural flexibility in defining group actions.