Problem 40 - Algebra: Pure and Applied

Question

Show that \(U(8)\) and \(U(12)\) are isomorphic.

Step-by-Step Solution

Answer

The groups \( U(8) \) and \( U(12) \) are isomorphic by mapping \( \{1, 3, 5, 7\} \) to \( \{1, 5, 7, 11\} \) with operations preserved.

Step 1: Define the Groups

The group of units modulo 8, denoted as \( U(8) \), is the set of integers less than 8 that are coprime to 8. The elements of \( U(8) \) are {1, 3, 5, 7}. Similarly, \( U(12) \) is the set of integers less than 12 that are coprime to 12, which is {1, 5, 7, 11}.

Step 2: Determine Group Order

Calculate the order of each group. Both \( U(8) \) and \( U(12) \) have 4 elements, since \( \varphi(8) = 4 \) and \( \varphi(12) = 4 \), where \( \varphi \) is Euler's totient function.

Step 3: Find Group Operation

Both groups, \( U(8) \) and \( U(12) \), use multiplication modulo 8 and 12, respectively. This means that the operation in both groups is multiplication followed by taking the remainder with 8 or 12, respectively.

Step 4: Explore Possible Isomorphisms

Find a bijective function \( \phi \) that maps elements of \( U(8) \) to \( U(12) \) such that multiplication operations correspond between elements of \( U(8) \) and \( U(12) \). One possible mapping is: \( \phi(1) = 1 \), \( \phi(3) = 5 \), \( \phi(5) = 7 \), and \( \phi(7) = 11 \).

Step 5: Verify Homomorphism Property

Check that the function \( \phi \) preserves the group operation. Verify that for every pair \( a, b \in U(8) \), \( \phi(a \cdot b) = \phi(a) \cdot \phi(b) \) in \( U(12) \). For example, \( 3 \cdot 5 \equiv 15 \equiv 7 \pmod{8} \), and \( 5 \cdot 7 \equiv 35 \equiv 11 \pmod{12} \). All cases of multiplication follow this preservation.

Step 6: Confirm Isomorphism

Since \( \phi \) is bijective (one-to-one and onto) and preserves the group operation, \( \phi \) is an isomorphism. Hence, \( U(8) \) and \( U(12) \) are isomorphic.

Key Concepts

Group Theory

Group theory is a fundamental concept in mathematics that studies the algebraic structures known as groups. A group is a set equipped with an operation that combines any two elements to form a third element while satisfying four key properties: closure, associativity, identity, and invertibility. In simpler terms, for a set to be considered a group:

• Closure: Applying the group operation to two elements in the set results in another element of the set.
• Associativity: The way the elements are grouped when performing the operation does not change the outcome.
• Identity Element: There exists an element in the group such that any element combined with it will return the same element.
• Inverse Elements: For each element, there is another element in the group that combines with it to result in the identity element.

When analyzing groups like \( U(8) \) and \( U(12) \), it's essential to verify these properties to understand their structures. These groups are also known as multiplicative groups of units modulo a number, where only numbers coprime to the modulus are included and form a group under multiplication modulo \( n \). Understanding how these properties work is central to examining group isomorphisms.

Euler's Totient Function

Euler's totient function, denoted \( \varphi(n) \), is a key concept in number theory. It counts the number of integers up to \( n \) that are coprime to \( n \), meaning they have no divisors other than 1 in common with \( n \). The function has crucial importance in group theory, particularly when discussing the group of units \( U(n) \).
For example, in the exercise given, we examine \( U(8) \) and \( U(12) \). Using Euler's totient function, we calculate:

• \( \varphi(8) \) = 4, as there are 4 numbers less than 8 and coprime to 8: \{1, 3, 5, 7\}.
• \( \varphi(12) \) = 4, as there are 4 numbers less than 12 and coprime to 12: \{1, 5, 7, 11\}.

The values of \( \varphi(8) \) and \( \varphi(12) \) are instrumental in determining the order of the groups \( U(8) \) and \( U(12) \), which is a critical step in proving their isomorphism. Recognizing that these groups have the same order is a preliminary step to establishing an isomorphism.

Cyclic Groups

Cyclic groups are a specific type of group where every element can be generated by repeatedly applying the group operation to any particular element, known as a generator. If a group is cyclic, there exists at least one element in the group that can produce every other element through this repeated operation. All cyclic groups are abelian, meaning the group operation is commutative.
When we explore groups such as \( U(8) \) or \( U(12) \), we are interested in determining whether they possess cyclicity. For a group to be cyclic, it must have one generator that can produce all other elements in the group. In our example, these groups aren't inherently cyclic with standard operations as the order of the groups does not align with properties of powers of any one element generating the entire group. However, cyclic groups play a significant role when finding isomorphic functions since they simplify mapping one group onto another if both are cyclic and have the same order, offering potential direct mapping relationships.

Modular Arithmetic

Modular arithmetic is often described as "clock arithmetic" because numbers wrap around after reaching a certain value, known as the modulus. In mathematical terms, two numbers are congruent modulo \( n \) if they have the same remainder when divided by \( n \). Modular arithmetic is pivotal when studying groups like \( U(8) \) and \( U(12) \).
In the context of these groups, the set of integers that are coprime to the given modulus are used with multiplication as the operation. This means:

• Each element of the group, when multiplied by another element, is followed by a division, and the remainder of this division is kept, yielding another element within the group—as required by the closure property.
• For instance, in \( U(8) \), multiplication follows modulo 8 rules, cycling through 8 before starting over, and likewise for \( U(12) \) with 12.

Understanding modular arithmetic is crucial for verifying and understanding the behavior of the functions used to prove isomorphisms, such as ensuring the consistency of the group operation across a mapping function. This consistency is foundational in establishing that, indeed, two groups are isomorphic.