Question

Find the index of \langle 10\rangle in \(\mathbb{Z}_{12}\).

Short answer

The index of \( \langle 10 \rangle \) in \( \mathbb{Z}_{12} \) is 2.

Step-by-step solution

Understanding the Problem

We need to find the index of the subgroup \( \langle 10 \rangle \) in the cyclic group \( \mathbb{Z}_{12} \). The notation \( \langle 10 \rangle \) represents the subgroup generated by the element \( 10 \). First, let's figure out what the subgroup \( \langle 10 \rangle \) consists of.

Generating the Subgroup

Calculate the elements of the subgroup \( \langle 10 \rangle \) in \( \mathbb{Z}_{12} \).ewline Compute: \( 10 \), \( 20 \equiv 8 \mod 12 \), \( 30 \equiv 6 \mod 12 \), \( 40 \equiv 4 \mod 12 \), \( 50 \equiv 2 \mod 12 \), \( 60 \equiv 0 \mod 12 \).ewline Thus, \( \langle 10 \rangle = \{0, 2, 4, 6, 8, 10\} \).

Counting Elements in the Subgroup

Count the number of elements in \( \langle 10 \rangle \). There are 6 elements: \{0, 2, 4, 6, 8, 10\}.

Finding the Order of the Group

Determine the order of \( \mathbb{Z}_{12} \), which is the number of elements in this group. \( \mathbb{Z}_{12} \) has 12 elements, which are \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}.

Calculating the Index

The index of a subgroup \( \langle a \rangle \) in a group \( G \) is the ratio of the order of \( G \) to the order of \( \langle a \rangle \). Therefore, the index of \( \langle 10 \rangle \) in \( \mathbb{Z}_{12} \) is \( \frac{12}{6} = 2 \).

Key concepts

Understanding Subgroup Index

The concept of the "subgroup index" can be understood as a way to measure how big a subgroup is compared to the group it belongs to. Suppose we have a group, like our cyclic group \( \mathbb{Z}_{12} \), which contains all integers from 0 to 11. If we choose a specific element from this group to generate a subgroup, this generated set will form a subgroup. The subgroup index is calculated by taking the size of the entire group and dividing it by the size of the subgroup. In mathematical terms, if \( G \) is a group and \( H \) is a subgroup, the index of \( H \) in \( G \) is expressed as \( |G : H| = \frac{|G|}{|H|} \). Here, \( |G| \) represents the order or size of the group, and \( |H| \) is the order of the subgroup. A lower index suggests that the subgroup is relatively large compared to the group, while a higher index indicates a smaller subgroup size compared to the group.

Basics of Modulo Operation

The modulo operation, often represented as \( \mod \), is a fundamental concept in number theory and group theory. When you perform a modulo operation, you are essentially finding the remainder of the division between two numbers. For example, in the expression \( a \mod n \), it calculates the remainder when \( a \) is divided by \( n \). This operation appears frequently in cyclic groups, such as \( \mathbb{Z}_{12} \), where every element is considered equivalent or congruent if they share the same remainder with respect to the modulus, which is 12 in this case. As applied in the exercise, where calculations like \( 30 \equiv 6 \mod 12 \) are involved, it helps us transform any integer into a corresponding number within the set \( \{0, 1, 2, \ldots, 11\} \). Useful properties of modulo include:

• \( (a + b) \mod n = [(a \mod n) + (b \mod n)] \mod n \)
• \( (a \times b) \mod n = [(a \mod n) \times (b \mod n)] \mod n \)

Understanding Group Order

In the realm of group theory, the "group order" simply refers to the total number of elements present in a group. It is one of the fundamental characteristics used to describe and analyze groups. In the exercise, the order of the group \( \mathbb{Z}_{12} \) is identified as 12, since it includes the integers from 0 to 11.Knowing the order of both a group and its subgroups can tell us a lot about the structure and properties of the group. The group order influences aspects like:

• Defining potential cycles within the group, particularly if it’s a cyclic group like \( \mathbb{Z}_{12} \).
• The potential size of any subgroups, since their orders must divide evenly into the group's order according to Lagrange's Theorem.

This exercise specifically explores the relationship between a subgroup's order and the entire group, and how their ratio gives the subgroup index. Understanding these concepts allows a deeper insight into the inner workings of mathematical structures and their symmetries.