Question

Find the order of the indicated element in the indicated quotient group. $$ 3+\langle 8\rangle \text { in } \mathbb{Z}_{12} /\langle 8\rangle $$

Short answer

The order of \( 3+\langle 8\rangle \) in \( \mathbb{Z}_{12} / \langle 8 \rangle \) is 4.

Step-by-step solution

Understand the Quotient Group

We are asked to find the order of an element in the quotient group \( \mathbb{Z}_{12} / \langle 8 \rangle \). In this quotient group, \( \langle 8 \rangle \) is the subgroup of \( \mathbb{Z}_{12} \). \( \langle 8 \rangle \) consists of all multiples of \( 8 \) under modulo \( 12 \).

Determine the Subgroup \(\langle 8\rangle\)

Calculate the subgroup \( \langle 8 \rangle \) in \( \mathbb{Z}_{12} \). Since \( \mathbb{Z}_{12} \) is a group under addition modulo \( 12 \), \( \langle 8 \rangle = \{ 0, 8 \} \) because \( 16 \equiv 4 \pmod{12} \), \( 24 \equiv 0 \pmod{12} \), and so on.

Find the Element in the Quotient Group

The element \( 3 + \langle 8 \rangle \) represents the coset \( \{ 3, 11 \} \) in the quotient group because you add \( 3 \) to every element of \( \langle 8 \rangle \). Therefore, \( 3 + \langle 8 \rangle \) is equivalent to \( \{ 3, 11 \} \).

Determine Order of the Element

The order of an element (coset) in a quotient group is the smallest positive integer \( n \) such that multiplying the element \( n \) times results in the identity element of the group. The identity in \( \mathbb{Z}_{12} / \langle 8 \rangle \) is \( \langle 8 \rangle \).

Check Multiples of the Element

Check the multiples of \( 3 + \langle 8 \rangle \) in the group:\( 1(3 + \langle 8 \rangle) = 3 + \langle 8 \rangle \), \( 2(3 + \langle 8 \rangle) = 6 + \langle 8 \rangle \), \( 3(3 + \langle 8 \rangle) = 9 + \langle 8 \rangle \), and \( 4(3 + \langle 8 \rangle) = 12 + \langle 8 \rangle = 0 + \langle 8 \rangle = \langle 8 \rangle \).

Conclusion on Order

Since \( 4(3 + \langle 8 \rangle) = \langle 8 \rangle \), the order of \( 3 + \langle 8 \rangle \) in \( \mathbb{Z}_{12} / \langle 8 \rangle \) is \( 4 \).

Key concepts

Order of an Element

The "order of an element" in a group is a fundamental concept in abstract algebra. It refers to the smallest positive number of times you must operate the element with itself to arrive back at the identity element (in additive groups, this is often zero). For an element represented by a coset in a quotient group, like \( 3 + \langle 8 \rangle \) in the group \( \mathbb{Z}_{12} / \langle 8 \rangle \), finding the order means calculating how many times we need to add this coset to itself to get \( \langle 8 \rangle \), the identity of the quotient group.In our original exercise, we found that multiplying or adding the element \( n \) times results in the identity when \( n=4 \). Therefore, the order of this particular element is 4. This concept highlights the repetitive pattern elements create as they cycle through group operations.

Cosets

Cosets are essential when working with quotient groups. The idea of a coset comes from dividing a group into equal sections using a subgroup. Given a subgroup, you can create a set by adding (or multiplying) every element of that subgroup with a particular element of the group. This set is called a "coset."For instance, in our exercise, \( 3 + \langle 8 \rangle \) forms the coset \( \{3, 11\} \). This means every element of the subgroup \( \langle 8 \rangle \) is added to 3. Cosets partition a group into disjoint sets, all of which either overlap entirely or not at all. They are particularly useful because they give a way to look at groups in terms of simpler, smaller building blocks, often simplifying complex problems.

Modular Arithmetic

Modular arithmetic is like playing with numbers that "wrap around" after they reach a certain value, called the modulus. It's sometimes referred to as "clock arithmetic" due to its cyclical nature. In the context of groups such as \( \mathbb{Z}_{12} \), operations are performed with a modulo of 12, meaning after reaching 12, you start counting back from zero.In this exercise, modular arithmetic explains why \( 8 \equiv 0 \pmod{12} \) within \( \mathbb{Z}_{12} \), and similarly why \( 16 \equiv 4 \pmod{12} \) and \( 24 \equiv 0 \pmod{12} \). Modulo operations allow for simplification in calculations, reducing problems to simpler ones that retain the essential characteristics of the original problem.

Subgroups

In group theory, a subgroup is any subset of a group that is itself a group, following the same operation as the original group. Subgroups must contain the identity element, and if any two elements of the subgroup are combined, their result must also be within the subgroup.In the exercise, \( \langle 8 \rangle \) is a subgroup of \( \mathbb{Z}_{12} \). It includes elements that are multiples of 8 under modular arithmetic with 12. Since \( \langle 8 \rangle = \{ 0, 8 \} \), it contains the identity element (0) and is closed under addition within the modulus. Understanding subgroups helps in constructing cosets and thus exploring the structure of quotient groups.