Finding the Multiplicative Group Generators for {1, -1, i, -i}

We are asked to find the generators of the multiplicative group consisting of the elements {1, -1, i, -i}. This is a fundamental concept in group theory, specifically dealing with cyclic groups and their Multiplicative Group Generators.

A cyclic group is a group that can be generated by a single element. This element, called a generator, is capable of producing every other element in the group through repeated application of the group operation (in this case, multiplication).

The given group is $G = \{1, -1, i, -i\}$ under multiplication. The identity element is 1. To find the generators, we need to check each element and see if its powers can produce all four elements of the group {1, -1, i, -i}. This process helps us identify the Multiplicative Group Generators.

Let's examine each element:

• Element 1: The powers of 1 under multiplication are $1^1=1$, $1^2=1$, $1^3=1$, and so on. The set generated by 1 is $\{1\}$. This does not include all elements of G, so 1 is not a generator.
• Element -1: The powers of -1 are:
  • $(-1)^1 = -1$
  • $(-1)^2 = (-1) \times (-1) = 1$
  • $(-1)^3 = (-1)^2 \times (-1) = 1 \times (-1) = -1$
  • $(-1)^4 = (-1)^3 \times (-1) = -1 \times (-1) = 1$
  The set generated by -1 is $\{-1, 1\}$. This does not include all elements of G, so -1 is not a generator.
• Element i: The powers of i are:
  • $i^1 = i$
  • $i^2 = -1$
  • $i^3 = i^2 \times i = -1 \times i = -i$
  • $i^4 = i^3 \times i = -i \times i = - (i^2) = -(-1) = 1$
  • $i^5 = i^4 \times i = 1 \times i = i$ (The powers repeat from here)
  The set generated by i is $\{i, -1, -i, 1\}$, which is exactly the set G. Therefore, i is a generator. This confirms i is one of the Multiplicative Group Generators.
• Element -i: The powers of -i are:
  • $(-i)^1 = -i$
  • $(-i)^2 = (-1)^2 \times i^2 = 1 \times (-1) = -1$
  • $(-i)^3 = (-i)^2 \times (-i) = -1 \times (-i) = i$
  • $(-i)^4 = (-i)^3 \times (-i) = i \times (-i) = -(i^2) = -(-1) = 1$
  • $(-i)^5 = (-i)^4 \times (-i) = 1 \times (-i) = -i$ (The powers repeat from here)
  The set generated by -i is $\{-i, -1, i, 1\}$, which is exactly the set G. Therefore, -i is also a generator.

Identifying the Generators using Group Theory Principles

Based on our calculations of powers, the elements that generate the entire multiplicative group {1, -1, i, -i} are i and -i. These are the Multiplicative Group Generators for this specific cyclic group. This analysis aligns with general group theory principles regarding cyclic groups and their generators.

For a cyclic group of order $n$, the number of generators is $\phi(n)$, where $\phi$ is Euler's totient function. In this group, the order is 4. Euler's totient function $\phi(4)$ counts the number of positive integers up to 4 that are relatively prime to 4. These are 1 and 3. So $\phi(4) = 2$. This tells us there should be exactly two generators for this cyclic group.

The orders of the elements are:

• Order of 1: 1 (since $1^1=1$)
• Order of -1: 2 (since $(-1)^2=1$ and $(-1)^1 \neq 1$)
• Order of i: 4 (since $i^4=1$ and $i^1, i^2, i^3 \neq 1$)
• Order of -i: 4 (since $(-i)^4=1$ and $(-i)^1, (-i)^2, (-i)^3 \neq 1$)

In a finite cyclic group of order $n$, an element $g$ is a generator if and only if its order is $n$. In our case, the group order is 4. The elements with order 4 are i and -i. Thus, i and -i are the Multiplicative Group Generators.

This detailed examination confirms that i and -i are the elements that can generate the entire group through multiplication, making them the Multiplicative Group Generators. This is a key concept in abstract algebra and group theory.