Question

Find the order of the indicated element in the indicated quotient group. $$ 2+\langle 6\rangle \text { in } \mathbb{Z}_{15} /\langle 6\rangle $$

Short answer

The order of \(2 + \langle 6 \rangle\) in \(\mathbb{Z}_{15}/\langle 6 \rangle\) is 3.

Step-by-step solution

Understand the Question

We need to find the order of the element presented as \(2 + \langle 6 \rangle\) in the quotient group \(\mathbb{Z}_{15} / \langle 6 \rangle\). This means we are looking for the smallest positive integer \(n\) such that \(n(2 + \langle 6 \rangle) = \langle 6 \rangle\) in \(\mathbb{Z}_{15} / \langle 6 \rangle\).

Define the Quotient Group Structure

The subgroup \(\langle 6 \rangle\) in \(\mathbb{Z}_{15}\) is generated by the integer 6, which consists of all multiples of 6: \( \{0, 6, 12\}\). The quotient group \(\mathbb{Z}_{15} / \langle 6 \rangle\) is the set of cosets: \(\{ \langle 6 \rangle, 1 + \langle 6 \rangle, 2 + \langle 6 \rangle, \ldots, 5 + \langle 6 \rangle \}\).

Apply the Order Condition

To find the order, compute \(n(2 + \langle 6 \rangle)\): - If \(n = 1\), \(1(2 + \langle 6 \rangle) = 2 + \langle 6 \rangle\)- If \(n = 2\), \(2(2 + \langle 6 \rangle) = 4 + \langle 6 \rangle\)- If \(n = 3\), \(3(2 + \langle 6 \rangle) = 6 + \langle 6 \rangle = 0 + \langle 6 \rangle\)Here, \(3(2 + \langle 6 \rangle)\) gives us \(\langle 6 \rangle\), showing that \(n = 3\) satisfies the condition.

Confirm the Result

The order of \(2 + \langle 6 \rangle\) is indeed 3 since \(3 \times 2 = 6\), and 6 is equivalent to 0 in \(\langle 6 \rangle\). Therefore, \(n(2 + \langle 6 \rangle) = \langle 6 \rangle\) when \(n = 3\).

Key concepts

Cosets

In group theory, cosets play a vital role in understanding the structure of more complex groups. A coset is formed by taking all the elements of a subgroup and adding a fixed element from the larger group to each of them.
For example, given a subgroup \( \langle 6 \rangle \) in the group \( \mathbb{Z}_{15} \), a coset \( 2 + \langle 6 \rangle \) would include all elements of \( \langle 6 \rangle \) added to 2, such as \( \{2, 8, 14\} \).
This allows us to partition the original group into distinct sets, each represented by a coset. Understanding these partitions helps us analyze the behavior and properties of quotient groups.

Integer Order

The integer order of an element in a group refers to the smallest positive integer \( n \) such that multiplying the element by \( n \) results in the identity element of the group. In the context of a quotient group, this concept helps establish a repetitive pattern or cycle.
In the problem, we need the smallest \( n \) so that multiplying \( 2 + \langle 6 \rangle \) by \( n \) produces the identity element, which in this case is \( \langle 6 \rangle \).
By checking successive products like \( 2 \times n + \langle 6 \rangle \), we identify that \( n = 3 \) yields \( \langle 6 \rangle \). Hence, the order of \( 2 + \langle 6 \rangle \) is 3. This discovery shows that three applications of the transformation yield the group identity.

Subgroup

A subgroup is essentially a subset of a group that maintains the group operation's properties within itself. That means any operation conducted within the subgroup results in an element that is still within the subgroup.
In this exercise, \( \langle 6 \rangle \) is a subgroup of \( \mathbb{Z}_{15} \). It consists of the elements \( \{0, 6, 12\} \) found by multiplying 6 with the elements in\( \mathbb{Z}_{15} \) under normal addition.
This subgroup acts as a building block in forming cosets which then are used to construct the more elaborate quotient group. Studying subgroups aids in grasping the internal structure and symmetry of groups.

Modular Arithmetic

Modular arithmetic, often referred to as clock arithmetic, is a system of arithmetic for integers where numbers "wrap around" after reaching a certain value known as the modulus.
In \( \mathbb{Z}_{15} \), calculations are done modulo 15. For instance, \( 17 \equiv 2 \pmod{15} \) since 17 minus 15 is 2. This wrapping around effect allows for a simplified form of calculation in many mathematical scenarios.
In our problem, applying modular arithmetic, we see that computing \( 6 + \langle 6 \rangle \) gives us \( 0 + \langle 6 \rangle \). Understanding such calculations underlies the determination of element order and assists in exploring deeper properties of groups like quotient groups.