Problem 37 - A Computational Introduction to Number Theory and Algebra

Question

Which of the following pairs of groups are isomorphic? Why or (d) \(\mathbb{Z}_{2} \times \mathbb{Z}\) and \(\mathbb{Z},(\mathrm{e})\) why not? (a) \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\) and \(\mathbb{Z}_{4},\) (b) \(\mathbb{Z}_{12}^{*}\) and \(\mathbb{Z}_{8}^{*},\) (c) \(\mathbb{Z}_{5}^{*}\) and \(\mathbb{Z}_{4}\), \(\mathbb{Q}\) and \(\mathbb{Z},\) (f) \(\mathbb{Z} \times \mathbb{Z}\) and \(\mathbb{Z}\)

Step-by-Step Solution

Answer

Question: Determine if the given pairs of groups are isomorphic or not: (a) \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\) and \(\mathbb{Z}_{4}\) (b) \(\mathbb{Z}_{12}^{*}\) and \(\mathbb{Z}_{8}^{*}\) (c) \(\mathbb{Z}_{5}^{*}\) and \(\mathbb{Z}_{4}\) (d) \(\mathbb{Z}_{2} \times \mathbb{Z}\) and \(\mathbb{Z}\) (e) \(\mathbb{Z} \times \mathbb{Z}\) and \(\mathbb{Z}\) (f) \(\mathbb{Q}\) and \(\mathbb{Z}\) Answer: (a) Isomorphic (b) Not isomorphic (c) Not isomorphic (d) Not isomorphic (e) Not isomorphic (f) Not isomorphic

Step 1: Determine the orders of the given groups

For each pair of groups, compute the order of each group. (a) \(|\mathbb{Z}_{2} \times \mathbb{Z}_{2}| = 2 \cdot 2 = 4\), and \(|\mathbb{Z}_{4}| = 4\) (b) \(|\mathbb{Z}_{12}^{*}| = 4\), and \(|\mathbb{Z}_{8}^{*}| = 4\) (counting the numbers relatively prime to 12 and 8) (c) \(|\mathbb{Z}_{5}^{*}| = 4\), and \(|\mathbb{Z}_{4}| = 4\) (d) \(|\mathbb{Z}_{2} \times \mathbb{Z}| = 2 \cdot \infty = \infty\), and \(|\mathbb{Z}| = \infty\) (e) \(|\mathbb{Z} \times \mathbb{Z}| = \infty \cdot \infty = \infty\), and \(|\mathbb{Z}| = \infty\) (f) \(|\mathbb{Q}| = \infty\), and \(|\mathbb{Z}| = \infty\)

Step 2: Check if the orders match

If the orders of the two groups are different, they cannot be isomorphic. In this case, all pairs of groups have the same orders.

Step 3: Provide an explanation

(a) The groups \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\) and \(\mathbb{Z}_{4}\) are isomorphic because they have the same order, and their respective elements exhibit the same cyclic behavior. (b) The groups \(\mathbb{Z}_{12}^{*}\) and \(\mathbb{Z}_{8}^{*}\) are not isomorphic because the group operation in \(\mathbb{Z}_{12}^{*}\) is multiplication modulo 12, while in \(\mathbb{Z}_{8}^{*}\) it is multiplication modulo 8. So, their respective elements do not exhibit the same behavior. (c) The groups \(\mathbb{Z}_{5}^{*}\) and \(\mathbb{Z}_{4}\) are not isomorphic. \(\mathbb{Z}_{5}^{*}\) is a multiplicative group modulo 5, while \(\mathbb{Z}_{4}\) is an additive group modulo 4. So, their respective elements do not exhibit the same behavior. (d) The groups \(\mathbb{Z}_{2} \times \mathbb{Z}\) and \(\mathbb{Z}\) are not isomorphic. \(\mathbb{Z}_{2} \times \mathbb{Z}\) contains elements of order 2, while \(\mathbb{Z}\) does not have any elements of order 2 (except for the identity element). (e) The groups \(\mathbb{Z} \times \mathbb{Z}\) and \(\mathbb{Z}\) are not isomorphic. \(\mathbb{Z} \times \mathbb{Z}\) is not cyclic, while \(\mathbb{Z}\) is cyclic. (f) The groups \(\mathbb{Q}\) and \(\mathbb{Z}\) are not isomorphic because \(\mathbb{Q}\) is a field and contains inverses for all non-zero elements under multiplication, while \(\mathbb{Z}\) does not have multiplicative inverses for all non-zero elements.

Key Concepts

Group Theory

In mathematics, group theory is the study of algebraic structures known as groups. A group is a set combined with an operation that satisfies four key properties: closure, associativity, identity, and invertibility.

• **Closure**: If you perform the group operation on any two elements in the group, you must get another element that is also in the group.
• **Associativity**: Changing the grouping of elements does not change the result of their operation (i.e., (\((a \cdot b) \cdot c = a \cdot (b \cdot c)\)).
• **Identity**: There must be an element in the group such that any element combined with it, under the group operation, returns that element itself.
• **Invertibility**: Every element must have an inverse within the group that, when combined with it under the group operation, yields the identity element.

The concept of isomorphism is crucial in group theory. Two groups are considered isomorphic if there is a one-to-one correspondence between their elements that preserves the group operation. This means that, in essence, the groups are structurally the same. Understanding these basic concepts helps students recognize why certain groups are isomorphic or not, such as in the given exercise.

Cyclic Groups

Cyclic groups are a special type of group where all elements can be generated by repeatedly applying the group operation to a single element, known as a generator.
These groups can be thought of as "spinning" around a central element. If you consider the set of integers under addition, the generator might be 1 or -1 because you can create any integer by adding or subtracting 1 repeatedly.

• A cyclic group with a finite number of elements is isomorphic to the integers modulo some positive integer \( n \).
• All elements in a cyclic group exhibit a repeating pattern.
• This property is crucial when considering whether two groups, such as \(\mathbb{Z}_{4}\) and \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\), are isomorphic. Both are cyclic and thus exhibit the same cyclical behavior.

Cyclic groups often appear in both theoretical and practical applications because they are simple to understand and model many natural processes.

Multiplicative Groups

Multiplicative groups are those in which the group operation is multiplication. These groups are essential in understanding number theory and cryptography.

• One example is \(\mathbb{Z}_{n}^{*}\), the multiplicative group of integers modulo \( n \). This group consists of integers that are coprime to \( n \) with multiplication being the operation.
• These groups are infinite if based on complex numbers but can be finite when considering fields or certain sets of integers.
• The structure of multiplicative groups can be somewhat different from additive groups (like \( \mathbb{Z}_{4} \)), so two groups with similar orders might still not be isomorphic due to different group operations.

In the problem set provided, an understanding of multiplicative versus additive behavior was necessary to determine why \(\mathbb{Z}_{5}^{*}\) and \(\mathbb{Z}_{4}\) are not isomorphic. This highlights the versatility and intricacy of group structures in mathematical theory.