Problem 41 - Algebra: Pure and Applied

Question

Show that in \(C^{*}\) the subgroup \(\langle i\rangle\) generated by \(i\) is isomorphic to \(\mathbb{Z}_{4}\)

Step-by-Step Solution

Answer

The subgroup \(\langle i \rangle\) is isomorphic to \(\mathbb{Z}_4\) due to their cyclic structure and element mapping.

Step 1: Understanding the Subgroup

The complex number system \(C^*\) consists of all non-zero complex numbers. The subgroup generated by \(i\) in \(C^*\) includes all powers of \(i\), i.e., \(\{i^n \mid n \in \mathbb{Z}\}\). The important observation here is the cyclic nature of powers of \(i\).

Step 2: Evaluating Powers of i

Calculate the powers of \(i\): - \(i^1 = i\) - \(i^2 = -1\) - \(i^3 = -i\) - \(i^4 = 1\) - Since \(i^4 = 1\), further powers will repeat this cycle, i.e., \(i^5 = i\), \(i^6 = -1\), and so on.

Step 3: Setting Up the Isomorphism

The subgroup \(\langle i \rangle = \{1, i, -1, -i\}\) functions like a cyclic group of order 4. Comparatively, \(\mathbb{Z}_4 = \{0, 1, 2, 3\}\) is also a cyclic group with order 4. Map the elements as \(i^0 \mapsto 0, i^1 \mapsto 1, i^2 \mapsto 2, i^3 \mapsto 3\). This correspondence maintains the group operation through addition modulo 4 and multiplication by powers of \(i\).

Step 4: Proving Isomorphism

To prove isomorphism, the mapping must be bijective and operation-preserving:1. **Bijective**: The map \(\phi: \{1, i, -1, -i\} \rightarrow \{0, 1, 2, 3\}\) is one-to-one and onto.2. **Operation-preserving**: \(\phi(i^a \cdot i^b) = \phi(i^a) + \phi(i^b) \mod 4\). The multiplication in \(\langle i \rangle\) corresponds to addition in \(\mathbb{Z}_4\). Verifying for each pair confirms the operation is preserved.

Key Concepts

Cyclic Groups

A cyclic group is a group that can be generated by a single element. This means all elements in the group can be expressed as powers of a particular element, known as the generator. For example, in the group \(Clangle i \rangle\), the generator is \(i\). The powers of \(i\) form a cyclic sequence: \(i^1 = i, i^2 = -1, i^3 = -i,\) and \(i^4 = 1\). After \(i^4\), the powers repeat their sequence.

• Cyclic groups can be finite or infinite.
• They are crucial in understanding the structure and behavior of different types of groups.
• The subgroup \(\langle i \rangle\) is a cyclic group, as it can be fully generated by repeated operations on one element, \(i\).

Cyclic groups, such as \(\mathbb{Z}_4\), have a clear and repetitive structure that makes them easier to study and apply in various areas of mathematics.

Powers of Complex Numbers

Complex numbers are numbers that include a real component and an imaginary component, represented as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.

• The imaginary unit \(i\) satisfies the equation \(i^2 = -1\).
• Powers of \(i\) cycle every four powers: \(i, -1, -i, \)and \(1\).

Calculating these powers:- \(i^1 = i\)- \(i^2 = -1\)- \(i^3 = -i\)- \(i^4 = 1\)This cyclical pattern is essential in understanding group behavior, particularly for cyclic groups like \(\langle i \rangle\). Each exponent corresponds to a finite set of possible outcomes, enabling a systematic approach to studying their properties.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value. This process helps to simplify calculations in cyclic groups, like \(\mathbb{Z}_4\), by restricting values to a set that repeats in a predictable manner.

• The arithmetic operates under the concept of modulus, denoted as \(%\).
• For example, in \(\mathbb{Z}_4\), any integer is equivalent to one of \(\{0, 1, 2, 3\}\) under modulo 4.

Consider: - \(5 \mod 4 = 1\)- \(6 \mod 4 = 2\)This means numbers "cycle" through these representatives, aligning well with the cycling of complex numbers' powers, thus maintaining the structural compatibility in group operations under isomorphisms, such as the one between \(\langle i \rangle\) and \(\mathbb{Z}_4\).

Complex Number System

The complex number system extends the real numbers by including elements that cannot be expressed solely as a real number. Complex numbers are represented in the form of \(a + bi\), where both \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

• They have applications in various fields like electrical engineering, quantum physics, and applied mathematics.
• The subset of non-zero complex numbers forms a group known as the multiplicative group of non-zero complex numbers, \(C^*\).

This group includes all complex numbers except zero, holding important properties like closure, associativity, the existence of an identity element (1), and the existence of inverse elements. In the context of the exercise, \(C^*\) helps highlight the structure and behavior of subgroups such as \(\langle i \rangle\), allowing for the exploration of isomorphisms and their equivalences to other well-known groups like \(\mathbb{Z}_4\). Understanding this system builds a foundational base for exploring higher mathematical concepts.